let f be FinSequence of (TOP-REAL 2); :: thesis: for G being Go-board st ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds
ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
let G be Go-board; :: thesis: ( ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) )
defpred S₁[ Element of NAT ] means for f being FinSequence of (TOP-REAL 2) st len f = $1 & ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds
ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g );
A1: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
let f be FinSequence of (TOP-REAL 2); :: thesis: ( len f = k + 1 & ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) )
assume that
A3: len f = k + 1 and
A4: for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) and
A5: f is one-to-one and
A6: f is unfolded and
A7: f is s.n.c. and
A8: f is special ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )

per cases ( k = 0 or k <> 0 ) ;

suppose A9: k = 0 ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )

take g = f; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
A10: dom f = {1} by A3, A9, FINSEQ_1:4, FINSEQ_1:def 3;

now

let n be Element of NAT ; :: thesis: ( n in dom g & n + 1 in dom g implies for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds
(abs (i1 - j1)) + (abs (i2 - j2)) = 1 )
assume that
A11: n in dom g and
A12: n + 1 in dom g ; :: thesis: for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds
(abs (i1 - j1)) + (abs (i2 - j2)) = 1
n = 1 by A10, A11, TARSKI:def 1;
hence for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds
(abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A10, A12, TARSKI:def 1; :: thesis: verum

end;

hence g is_sequence_on G by A4, GOBOARD1:def 11; :: thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
thus ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) by A5, A6, A7, A8; :: thesis: verum

end;

suppose A13: k <> 0 ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )

A14: len (f | k) = k by A3, FINSEQ_1:80, NAT_1:11;
set f1 = f | k;
A15: f | k is unfolded by A3, A6, A13, Lm1;
A16: f | k is s.n.c. by A7, GOBOARD2:12;
f | k = f | (Seg k) by FINSEQ_1:def 15;
then A17: f | k is one-to-one by A5, FUNCT_1:84;
A18: dom G = Seg (len G) by FINSEQ_1:def 3;
1 <= len f by A3, NAT_1:11;
then A19: k + 1 in dom f by A3, FINSEQ_3:27;
then consider j1, j2 being Element of NAT such that
A20: [j1,j2] in Indices G and
A21: f /. (k + 1) = G * (j1,j2) by A4;
A22: Indices G = [:(dom G),(Seg (width G)):] by MATRIX_1:def 5;
then A23: j1 in dom G by A20, ZFMISC_1:106;
A24: 0 + 1 <= k by A13, NAT_1:13;
then A25: 1 in Seg k by FINSEQ_1:3;
A26: k <= k + 1 by NAT_1:11;
then A27: k in dom f by A3, A24, FINSEQ_3:27;
then consider i1, i2 being Element of NAT such that
A28: [i1,i2] in Indices G and
A29: f /. k = G * (i1,i2) by A4;
reconsider l1 = Line (G,i1), c1 = Col (G,i2) as FinSequence of (TOP-REAL 2) ;
set x1 = X_axis l1;
set y1 = Y_axis l1;
set x2 = X_axis c1;
set y2 = Y_axis c1;
A30: ( dom (Y_axis l1) = Seg (len (Y_axis l1)) & len (Y_axis l1) = len l1 ) by FINSEQ_1:def 3, GOBOARD1:def 4;
A31: dom (f | k) = Seg (len (f | k)) by FINSEQ_1:def 3;
A32: f | k is special

proof

let n be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= n or not n + 1 <= len (f | k) or ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 )
assume that
A33: 1 <= n and
A34: n + 1 <= len (f | k) ; :: thesis: ( ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 )
n + 1 in dom (f | k) by A33, A34, SEQ_4:151;
then A35: (f | k) /. (n + 1) = f /. (n + 1) by A27, A14, A31, FINSEQ_4:86;
len (f | k) <= len f by A3, A26, FINSEQ_1:80;
then A36: n + 1 <= len f by A34, XXREAL_0:2;
n in dom (f | k) by A33, A34, SEQ_4:151;
then (f | k) /. n = f /. n by A27, A14, A31, FINSEQ_4:86;
hence ( ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 ) by A8, A33, A35, A36, TOPREAL1:def 7; :: thesis: verum

end;

now

let n be Element of NAT ; :: thesis: ( n in dom (f | k) implies ex i, j being Element of NAT st
( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) )
assume A37: n in dom (f | k) ; :: thesis: ex i, j being Element of NAT st
( [i,j] in Indices G & (f | k) /. n = G * (i,j) )
then n in dom f by A27, A14, A31, FINSEQ_4:86;
then consider i, j being Element of NAT such that
A38: ( [i,j] in Indices G & f /. n = G * (i,j) ) by A4;
take i = i; :: thesis: ex j being Element of NAT st
( [i,j] in Indices G & (f | k) /. n = G * (i,j) )
take j = j; :: thesis: ( [i,j] in Indices G & (f | k) /. n = G * (i,j) )
thus ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) by A27, A14, A31, A37, A38, FINSEQ_4:86; :: thesis: verum

end;

then consider g1 being FinSequence of (TOP-REAL 2) such that
A39: g1 is_sequence_on G and
A40: g1 is one-to-one and
A41: g1 is unfolded and
A42: g1 is s.n.c. and
A43: g1 is special and
A44: L~ g1 = L~ (f | k) and
A45: g1 /. 1 = (f | k) /. 1 and
A46: g1 /. (len g1) = (f | k) /. (len (f | k)) and
A47: len (f | k) <= len g1 by A2, A14, A17, A15, A16, A32;
A48: for n being Element of NAT st n in dom g1 & n + 1 in dom g1 holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g1 /. n = G * (m,k) & g1 /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A39, GOBOARD1:def 11;
A49: ( 1 < k implies rng g1 c= L~ (f | k) )

proof

assume 1 < k ; :: thesis: rng g1 c= L~ (f | k)
then A50: 1 + 1 <= k by NAT_1:13;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g1 or x in L~ (f | k) )
assume x in rng g1 ; :: thesis: x in L~ (f | k)
then ex n being Element of NAT st
( n in dom g1 & g1 /. n = x ) by PARTFUN2:4;
hence x in L~ (f | k) by A14, A44, A47, A50, GOBOARD1:16, XXREAL_0:2; :: thesis: verum

end;

A51: k in Seg k by A24, FINSEQ_1:3;
A52: ( k = 1 implies ( L~ g1 = {} & rng g1 = {(f /. k)} ) )

proof

A53: g1 /. (len g1) = f /. k by A27, A14, A51, A46, FINSEQ_4:86;
assume A54: k = 1 ; :: thesis: ( L~ g1 = {} & rng g1 = {(f /. k)} )
hence L~ g1 = {} by A14, A44, TOPREAL1:28; :: thesis: rng g1 = {(f /. k)}
then A55: ( len g1 = 1 or len g1 = 0 ) by TOPREAL1:28;
A56: rng g1 c= {(f /. k)}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g1 or x in {(f /. k)} )
assume x in rng g1 ; :: thesis: x in {(f /. k)}
then consider n being Element of NAT such that
A57: n in dom g1 and
A58: g1 /. n = x by PARTFUN2:4;
n in Seg (len g1) by A57, FINSEQ_1:def 3;
then n = len g1 by A55, FINSEQ_1:4, TARSKI:def 1;
hence x in {(f /. k)} by A53, A58, TARSKI:def 1; :: thesis: verum

end;

1 <= len g1 by A3, A47, A54, FINSEQ_1:80;
then len g1 in dom g1 by FINSEQ_3:27;
then f /. k in rng g1 by A53, PARTFUN2:4;
then {(f /. k)} c= rng g1 by ZFMISC_1:37;
hence rng g1 = {(f /. k)} by A56, XBOOLE_0:def 10; :: thesis: verum

end;

A59: len c1 = len G by MATRIX_1:def 9;
then A60: dom c1 = Seg (len G) by FINSEQ_1:def 3
.= dom G by FINSEQ_1:def 3 ;
A61: ( dom (Y_axis c1) = Seg (len (Y_axis c1)) & len (Y_axis c1) = len c1 ) by FINSEQ_1:def 3, GOBOARD1:def 4;
A62: ( dom (X_axis l1) = Seg (len (X_axis l1)) & len (X_axis l1) = len l1 ) by FINSEQ_1:def 3, GOBOARD1:def 3;
A63: dom (X_axis c1) = Seg (len (X_axis c1)) by FINSEQ_1:def 3;
A64: len (X_axis c1) = len c1 by GOBOARD1:def 3;
then A65: dom c1 = Seg (len (X_axis c1)) by FINSEQ_1:def 3
.= dom (X_axis c1) by FINSEQ_1:def 3 ;
A66: i1 in dom G by A28, A22, ZFMISC_1:106;
then A67: X_axis l1 is constant by GOBOARD1:def 6;
A68: i2 in Seg (width G) by A28, A22, ZFMISC_1:106;
then A69: X_axis c1 is increasing by GOBOARD1:def 9;
A70: Y_axis c1 is constant by A68, GOBOARD1:def 7;
A71: Y_axis l1 is increasing by A66, GOBOARD1:def 8;
A72: len l1 = width G by MATRIX_1:def 8;
then A73: Seg (width G) = dom l1 by FINSEQ_1:def 3;
A74: j2 in Seg (width G) by A20, A22, ZFMISC_1:106;
A75: for n being Element of NAT st n in dom g1 holds
ex m, k being Element of NAT st
( [m,k] in Indices G & g1 /. n = G * (m,k) ) by A39, GOBOARD1:def 11;

now

per cases ( i1 = j1 or i2 = j2 ) by A8, A27, A28, A29, A19, A20, A21, GOBOARD2:16;

suppose A76: i1 = j1 ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )

set ppi = G * (i1,i2);
set pj = G * (i1,j2);

now

per cases ( i2 > j2 or i2 = j2 or i2 < j2 ) by XXREAL_0:1;

case A77: i2 > j2 ; :: thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

l1 /. i2 = l1 . i2 by A68, A73, PARTFUN1:def 8;
then A78: l1 /. i2 = G * (i1,i2) by A68, MATRIX_1:def 8;
then A79: (Y_axis l1) . i2 = (G * (i1,i2)) `2 by A68, A30, A72, GOBOARD1:def 4;
l1 /. j2 = l1 . j2 by A74, A73, PARTFUN1:def 8;
then A80: l1 /. j2 = G * (i1,j2) by A74, MATRIX_1:def 8;
then A81: (Y_axis l1) . j2 = (G * (i1,j2)) `2 by A74, A30, A72, GOBOARD1:def 4;
then A82: (G * (i1,j2)) `2 < (G * (i1,i2)) `2 by A68, A74, A71, A30, A72, A77, A79, SEQM_3:def 1;
reconsider l = i2 - j2 as Element of NAT by A77, INT_1:18;
defpred S₂[ Nat, set ] means for m being Element of NAT st m = i2 - $1 holds
$2 = G * (i1,m);
set lk = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } ;
A83: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:57;

A84: now

let n be Element of NAT ; :: thesis: ( n in Seg l implies ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) ) )
assume n in Seg l ; :: thesis: ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) )
then A85: n <= l by FINSEQ_1:3;
l <= i2 by XREAL_1:45;
then reconsider w = i2 - n as Element of NAT by A85, INT_1:18, XXREAL_0:2;
( i2 - n <= i2 & i2 <= width G ) by A68, FINSEQ_1:3, XREAL_1:45;
then A86: w <= width G by XXREAL_0:2;
A87: 1 <= j2 by A74, FINSEQ_1:3;
i2 - l <= i2 - n by A85, XREAL_1:15;
then 1 <= w by A87, XXREAL_0:2;
then w in Seg (width G) by A86, FINSEQ_1:3;
hence ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) ) by A22, A66, ZFMISC_1:106; :: thesis: verum

end;

A88: now

let n be Nat; :: thesis: ( n in Seg l implies ex p being Element of the U1 of (TOP-REAL 2) st S₂[n,p] )
assume n in Seg l ; :: thesis: ex p being Element of the U1 of (TOP-REAL 2) st S₂[n,p]
then reconsider m = i2 - n as Element of NAT by A84;
take p = G * (i1,m); :: thesis: S₂[n,p]
thus S₂[n,p] ; :: thesis: verum

end;

consider g2 being FinSequence of (TOP-REAL 2) such that
A89: ( len g2 = l & ( for n being Nat st n in Seg l holds
S₂[n,g2 /. n] ) ) from FINSEQ_4:sch 1(A88);
take g = g1 ^ g2; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A90: dom g2 = Seg l by A89, FINSEQ_1:def 3;

now

let n be Element of NAT ; :: thesis: ( n in dom g2 implies ex k, m being Element of NAT st
( [k,m] in Indices G & g2 /. n = G * (k,m) ) )
assume A91: n in dom g2 ; :: thesis: ex k, m being Element of NAT st
( [k,m] in Indices G & g2 /. n = G * (k,m) )
then reconsider m = i2 - n as Element of NAT by A84, A90;
take k = i1; :: thesis: ex m being Element of NAT st
( [k,m] in Indices G & g2 /. n = G * (k,m) )
take m = m; :: thesis: ( [k,m] in Indices G & g2 /. n = G * (k,m) )
thus ( [k,m] in Indices G & g2 /. n = G * (k,m) ) by A84, A89, A90, A91; :: thesis: verum

end;

then A92: for n being Element of NAT st n in dom g holds
ex i, j being Element of NAT st
( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:39;
A93: dom g2 = Seg (len g2) by FINSEQ_1:def 3;
A94: (X_axis l1) . i2 = (G * (i1,i2)) `1 by A68, A62, A72, A78, GOBOARD1:def 3;

A95: now

let n be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds
( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 )
let p be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & g2 /. n = p implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 ) )
assume that
A96: n in dom g2 and
A97: g2 /. n = p ; :: thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 )
reconsider n1 = i2 - n as Element of NAT by A84, A90, A96;
n <= len g2 by A96, FINSEQ_3:27;
then A98: i2 - (len g2) <= n1 by XREAL_1:15;
set pn = G * (i1,n1);
A99: g2 /. n = G * (i1,n1) by A89, A93, A96;
A100: i2 - n in Seg (width G) by A84, A89, A93, A96;
then A101: (X_axis l1) . n1 = (X_axis l1) . i2 by A68, A67, A62, A72, SEQM_3:def 15;
l1 /. n1 = l1 . n1 by A73, A100, PARTFUN1:def 8;
then A102: l1 /. n1 = G * (i1,n1) by A100, MATRIX_1:def 8;
then A103: (Y_axis l1) . n1 = (G * (i1,n1)) `2 by A30, A72, A100, GOBOARD1:def 4;
(X_axis l1) . n1 = (G * (i1,n1)) `1 by A62, A72, A100, A102, GOBOARD1:def 3;
hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 ) by A68, A74, A71, A30, A72, A89, A79, A81, A94, A97, A100, A99, A98, A101, A103, SEQ_4:154, XREAL_1:45; :: thesis: ( p in rng l1 & p `2 < (G * (i1,i2)) `2 )
dom l1 = Seg (len l1) by FINSEQ_1:def 3;
hence p in rng l1 by A72, A97, A100, A99, A102, PARTFUN2:4; :: thesis: p `2 < (G * (i1,i2)) `2
1 <= n by A96, FINSEQ_3:27;
then n1 < i2 by XREAL_1:46;
hence p `2 < (G * (i1,i2)) `2 by A68, A71, A30, A72, A79, A97, A100, A99, A103, SEQM_3:def 1; :: thesis: verum

end;

A104: g2 is special

proof

let n be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
set p = g2 /. n;
assume A105: ( 1 <= n & n + 1 <= len g2 ) ; :: thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
then n in dom g2 by SEQ_4:151;
then A106: (g2 /. n) `1 = (G * (i1,i2)) `1 by A95;
n + 1 in dom g2 by A105, SEQ_4:151;
hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A95, A106; :: thesis: verum

end;

A107: now

let n, m be Element of NAT ; :: thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds
q `2 < p `2
let p, q be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies q `2 < p `2 )
assume that
A108: n in dom g2 and
A109: m in dom g2 and
A110: n < m and
A111: ( g2 /. n = p & g2 /. m = q ) ; :: thesis: q `2 < p `2
A112: i2 - n in Seg (width G) by A84, A90, A108;
reconsider n1 = i2 - n, m1 = i2 - m as Element of NAT by A84, A90, A108, A109;
set pn = G * (i1,n1);
set pm = G * (i1,m1);
A113: m1 < n1 by A110, XREAL_1:17;
l1 /. n1 = l1 . n1 by A73, A84, A90, A108, PARTFUN1:def 8;
then l1 /. n1 = G * (i1,n1) by A112, MATRIX_1:def 8;
then A114: (Y_axis l1) . n1 = (G * (i1,n1)) `2 by A30, A72, A112, GOBOARD1:def 4;
A115: i2 - m in Seg (width G) by A84, A90, A109;
l1 /. m1 = l1 . m1 by A73, A84, A90, A109, PARTFUN1:def 8;
then l1 /. m1 = G * (i1,m1) by A115, MATRIX_1:def 8;
then A116: (Y_axis l1) . m1 = (G * (i1,m1)) `2 by A30, A72, A115, GOBOARD1:def 4;
( g2 /. n = G * (i1,n1) & g2 /. m = G * (i1,m1) ) by A89, A90, A108, A109;
hence q `2 < p `2 by A71, A30, A72, A111, A112, A115, A113, A114, A116, SEQM_3:def 1; :: thesis: verum

end;

for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds
LSeg (g2,n) misses LSeg (g2,m)

proof

let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) )
assume that
A117: m > n + 1 and
A118: n in dom g2 and
A119: n + 1 in dom g2 and
A120: m in dom g2 and
A121: m + 1 in dom g2 and
A122: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A123: ( p1 `1 = (G * (i1,i2)) `1 & p2 `1 = (G * (i1,i2)) `1 ) by A95, A118, A119;
n < n + 1 by NAT_1:13;
then A124: p2 `2 < p1 `2 by A107, A118, A119;
set lp = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p2 `2 <= w `2 & w `2 <= p1 `2 ) } ;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } ;
A125: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:57;
m < m + 1 by NAT_1:13;
then A126: q2 `2 < q1 `2 by A107, A120, A121;
A127: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
consider x being Element of (LSeg (g2,n)) /\ (LSeg (g2,m));
A128: x in LSeg (g2,n) by A122, XBOOLE_0:def 4;
A129: ( q1 `1 = (G * (i1,i2)) `1 & q2 `1 = (G * (i1,i2)) `1 ) by A95, A120, A121;
A130: x in LSeg (g2,m) by A122, XBOOLE_0:def 4;
( 1 <= m & m + 1 <= len g2 ) by A120, A121, FINSEQ_3:27;
then LSeg (g2,m) = LSeg (q2,q1) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } by A126, A129, A127, TOPREAL3:15 ;
then A131: ex tm being Point of (TOP-REAL 2) st
( tm = x & tm `1 = (G * (i1,i2)) `1 & q2 `2 <= tm `2 & tm `2 <= q1 `2 ) by A130;
( 1 <= n & n + 1 <= len g2 ) by A118, A119, FINSEQ_3:27;
then LSeg (g2,n) = LSeg (p2,p1) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p2 `2 <= w `2 & w `2 <= p1 `2 ) } by A124, A123, A125, TOPREAL3:15 ;
then A132: ex tn being Point of (TOP-REAL 2) st
( tn = x & tn `1 = (G * (i1,i2)) `1 & p2 `2 <= tn `2 & tn `2 <= p1 `2 ) by A128;
q1 `2 < p2 `2 by A107, A117, A119, A120;
hence contradiction by A132, A131, XXREAL_0:2; :: thesis: verum

end;

then A133: g2 is s.n.c. by GOBOARD2:6;
A134: not f /. k in L~ g2

proof

set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume f /. k in L~ g2 ; :: thesis: contradiction
then consider X being set such that
A135: f /. k in X and
A136: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A137: X = LSeg (g2,m) and
A138: ( 1 <= m & m + 1 <= len g2 ) by A136;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A139: m in dom g2 by A138, SEQ_4:151;
then A140: q1 `1 = (G * (i1,i2)) `1 by A95;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } ;
A141: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
A142: m + 1 in dom g2 by A138, SEQ_4:151;
then A143: q2 `1 = (G * (i1,i2)) `1 by A95;
m < m + 1 by NAT_1:13;
then A144: q2 `2 < q1 `2 by A107, A139, A142;
LSeg (g2,m) = LSeg (q2,q1) by A138, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } by A140, A143, A144, A141, TOPREAL3:15 ;
then ex p being Point of (TOP-REAL 2) st
( p = f /. k & p `1 = (G * (i1,i2)) `1 & q2 `2 <= p `2 & p `2 <= q1 `2 ) by A135, A137;
hence contradiction by A29, A95, A139; :: thesis: verum

end;

(X_axis l1) . j2 = (G * (i1,j2)) `1 by A74, A62, A72, A80, GOBOARD1:def 3;
then A145: (G * (i1,i2)) `1 = (G * (i1,j2)) `1 by A68, A74, A67, A62, A72, A94, SEQM_3:def 15;

now

let n, m be Element of NAT ; :: thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m )
assume that
A146: ( n in dom g2 & m in dom g2 ) and
A147: n <> m ; :: thesis: not g2 /. n = g2 /. m
reconsider n1 = i2 - n, m1 = i2 - m as Element of NAT by A84, A90, A146;
A148: ( g2 /. n = G * (i1,n1) & g2 /. m = G * (i1,m1) ) by A89, A90, A146;
assume A149: g2 /. n = g2 /. m ; :: thesis: contradiction
( [i1,(i2 - n)] in Indices G & [i1,(i2 - m)] in Indices G ) by A84, A90, A146;
then n1 = m1 by A148, A149, GOBOARD1:21;
hence contradiction by A147; :: thesis: verum

end;

then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds
n = m ;
then A150: g2 is one-to-one by PARTFUN2:16;
reconsider m1 = i2 - l as Element of NAT ;
A151: G * (i1,j2) = |[((G * (i1,j2)) `1),((G * (i1,j2)) `2)]| by EUCLID:57;
A152: LSeg (f,k) = LSeg ((G * (i1,j2)),(G * (i1,i2))) by A3, A24, A29, A21, A76, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A82, A145, A83, A151, TOPREAL3:15 ;
A153: rng g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g2 or x in LSeg (f,k) )
assume x in rng g2 ; :: thesis: x in LSeg (f,k)
then consider n being Element of NAT such that
A154: n in dom g2 and
A155: g2 /. n = x by PARTFUN2:4;
reconsider n1 = i2 - n as Element of NAT by A84, A89, A93, A154;
set pn = G * (i1,n1);
A156: g2 /. n = G * (i1,n1) by A89, A93, A154;
then A157: (G * (i1,n1)) `2 <= (G * (i1,i2)) `2 by A95, A154;
( (G * (i1,n1)) `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= (G * (i1,n1)) `2 ) by A95, A154, A156;
hence x in LSeg (f,k) by A152, A155, A156, A157; :: thesis: verum

end;

A158: now

let n be Element of NAT ; :: thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A159: n in dom g2 and
A160: n + 1 in dom g2 ; :: thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1
reconsider m1 = i2 - n, m2 = i2 - (n + 1) as Element of NAT by A84, A90, A159, A160;
let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A161: [l1,l2] in Indices G and
A162: [l3,l4] in Indices G and
A163: g2 /. n = G * (l1,l2) and
A164: g2 /. (n + 1) = G * (l3,l4) ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( [i1,(i2 - (n + 1))] in Indices G & g2 /. (n + 1) = G * (i1,m2) ) by A84, A89, A90, A160;
then A165: ( l3 = i1 & l4 = m2 ) by A162, A164, GOBOARD1:21;
( [i1,(i2 - n)] in Indices G & g2 /. n = G * (i1,m1) ) by A84, A89, A90, A159;
then ( l1 = i1 & l2 = m1 ) by A161, A163, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs ((i2 - n) - (i2 - (n + 1)))) by A165, ABSVALUE:7
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

now

let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A166: [l1,l2] in Indices G and
A167: [l3,l4] in Indices G and
A168: g1 /. (len g1) = G * (l1,l2) and
A169: g2 /. 1 = G * (l3,l4) and
len g1 in dom g1 and
A170: 1 in dom g2 ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
reconsider m1 = i2 - 1 as Element of NAT by A84, A90, A170;
( [i1,(i2 - 1)] in Indices G & g2 /. 1 = G * (i1,m1) ) by A84, A89, A90, A170;
then A171: ( l3 = i1 & l4 = m1 ) by A167, A169, GOBOARD1:21;
(f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:86;
then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A166, A168, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs (i2 - (i2 - 1))) by A171, ABSVALUE:7
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

then for n being Element of NAT st n in dom g & n + 1 in dom g holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A48, A158, GOBOARD1:40;
hence g is_sequence_on G by A92, GOBOARD1:def 11; :: thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A172: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def 5;
A173: not f /. k in rng g2

proof

assume f /. k in rng g2 ; :: thesis: contradiction
then consider n being Element of NAT such that
A174: n in dom g2 and
A175: g2 /. n = f /. k by PARTFUN2:4;
reconsider n1 = i2 - n as Element of NAT by A84, A89, A93, A174;
( [i1,(i2 - n)] in Indices G & g2 /. n = G * (i1,n1) ) by A84, A89, A93, A174;
then A176: n1 = i2 by A28, A29, A175, GOBOARD1:21;
0 < n by A93, A174, FINSEQ_1:3;
hence contradiction by A176; :: thesis: verum

end;

(rng g1) /\ (rng g2) = {}

proof

consider x being Element of (rng g1) /\ (rng g2);
assume A177: not (rng g1) /\ (rng g2) = {} ; :: thesis: contradiction
then A178: x in rng g2 by XBOOLE_0:def 4;
A179: x in rng g1 by A177, XBOOLE_0:def 4;

now

per cases ( k = 1 or 1 < k ) by A24, XXREAL_0:1;

suppose k = 1 ; :: thesis: contradiction

hence contradiction by A52, A173, A179, A178, TARSKI:def 1; :: thesis: verum

end;

suppose 1 < k ; :: thesis: contradiction

then ( x in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A153, A179, A178, GOBOARD2:9, XBOOLE_0:def 4;
hence contradiction by A173, A178, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

then rng g1 misses rng g2 by XBOOLE_0:def 7;
hence g is one-to-one by A40, A150, FINSEQ_3:98; :: thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A180: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds
(LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}

proof

let n be Element of NAT ; :: thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} )
assume that
A181: 1 <= n and
A182: n + 2 <= len g2 ; :: thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}
A183: n + 1 in dom g2 by A181, A182, SEQ_4:152;
then g2 /. (n + 1) in rng g2 by PARTFUN2:4;
then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u1 being Point of (TOP-REAL 2) such that
A184: g2 /. (n + 1) = u1 and
A185: u1 `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u1 `2 and
u1 `2 <= (G * (i1,i2)) `2 ;
A186: n + 2 in dom g2 by A181, A182, SEQ_4:152;
then g2 /. (n + 2) in rng g2 by PARTFUN2:4;
then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u2 being Point of (TOP-REAL 2) such that
A187: g2 /. (n + 2) = u2 and
A188: u2 `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u2 `2 and
u2 `2 <= (G * (i1,i2)) `2 ;
( n + (1 + 1) = (n + 1) + 1 & 1 <= n + 1 ) by NAT_1:11;
then A189: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A182, A184, A187, TOPREAL1:def 5;
n + 1 < (n + 1) + 1 by NAT_1:13;
then A190: u2 `2 < u1 `2 by A107, A183, A186, A184, A187;
A191: n in dom g2 by A181, A182, SEQ_4:152;
then g2 /. n in rng g2 by PARTFUN2:4;
then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u being Point of (TOP-REAL 2) such that
A192: g2 /. n = u and
A193: u `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u `2 and
u `2 <= (G * (i1,i2)) `2 ;
n + 1 <= n + 2 by XREAL_1:8;
then n + 1 <= len g2 by A182, XXREAL_0:2;
then A194: LSeg (g2,n) = LSeg (u,u1) by A181, A192, A184, TOPREAL1:def 5;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } ;
n < n + 1 by NAT_1:13;
then A195: u1 `2 < u `2 by A107, A191, A183, A192, A184;
then A196: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } by A185, A190;
( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:57;
then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } by A192, A193, A187, A188, A190, A195, TOPREAL3:15, XXREAL_0:2;
hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A192, A184, A187, A194, A189, A196, TOPREAL1:14; :: thesis: verum

end;

thus g is unfolded :: thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

let n be Nat; :: according to TOPREAL1:def 8 :: thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} )
assume that
A197: 1 <= n and
A198: n + 2 <= len g ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}
A199: (n + 1) + 1 <= len g by A198;
A200: n + (1 + 1) = (n + 1) + 1 ;
A201: n <= n + 1 by NAT_1:11;
n + 1 <= (n + 1) + 1 by NAT_1:11;
then A202: n + 1 <= len g by A198, XXREAL_0:2;
A203: len g = (len g1) + (len g2) by FINSEQ_1:35;
(n + 2) - (len g1) = (n - (len g1)) + 2 ;
then A204: (n - (len g1)) + 2 <= len g2 by A198, A203, XREAL_1:22;
A205: ( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11;

per cases ( n + 2 <= len g1 or len g1 < n + 2 ) ;

suppose A206: n + 2 <= len g1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

A207: n + (1 + 1) = (n + 1) + 1 ;
A208: n + 1 in dom g1 by A197, A206, SEQ_4:152;
then A209: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:83;
n in dom g1 by A197, A206, SEQ_4:152;
then A210: LSeg (g1,n) = LSeg (g,n) by A208, TOPREAL3:25;
n + 2 in dom g1 by A197, A206, SEQ_4:152;
then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A208, A207, TOPREAL3:25;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A197, A206, A210, A209, TOPREAL1:def 8; :: thesis: verum

end;

suppose len g1 < n + 2 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then (len g1) + 1 <= n + 2 by NAT_1:13;
then A211: len g1 <= (n + 2) - 1 by XREAL_1:21;

now

per cases ( len g1 = n + 1 or len g1 <> n + 1 ) ;

suppose A212: len g1 = n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

now

1 < len g1 by A197, A212, NAT_1:13;
then A213: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A213, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A214: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A197, A198, A202, A205, TOPREAL1:27;
then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def 4;
then A215: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:37;
A216: 1 <= (len g) - (len g1) by A199, A212, XREAL_1:21;
then 1 in dom g2 by A203, FINSEQ_3:27;
then A217: g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u1 being Point of (TOP-REAL 2) such that
A218: g2 /. 1 = u1 and
u1 `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u1 `2 and
u1 `2 <= (G * (i1,i2)) `2 ;
G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (i1,j2))) by RLTOPSP1:69;
then A219: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A172, A153, A217, A218, TOPREAL1:12;
1 <= n + 1 by NAT_1:11;
then A220: n + 1 in dom g1 by A212, FINSEQ_3:27;
then A221: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A212, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
n in dom g1 by A197, A201, A212, FINSEQ_3:27;
then A222: LSeg (g,n) = LSeg (g1,n) by A220, TOPREAL3:25;
g /. (n + 2) = g2 /. 1 by A200, A203, A212, A216, SEQ_4:153;
then A223: LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A198, A205, A221, A218, TOPREAL1:def 5;
LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:26;
then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A214, A222, A221, A219, A223, XBOOLE_1:27;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A215, XBOOLE_0:def 10; :: thesis: verum

end;

suppose len g1 <> n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then len g1 < n + 1 by A211, XXREAL_0:1;
then A224: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;

now

per cases ( len g1 = n or len g1 <> n ) ;

suppose A225: len g1 = n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then 1 <= len g2 by A202, A203, XREAL_1:8;
then A226: g /. (n + 1) = g2 /. 1 by A225, SEQ_4:153;
A227: 0 + 2 <= len g2 by A198, A203, A225, XREAL_1:8;
then 1 <= len g2 by XXREAL_0:2;
then A228: 1 in dom g2 by FINSEQ_3:27;
then g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u1 being Point of (TOP-REAL 2) such that
A229: g2 /. 1 = u1 and
A230: u1 `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u1 `2 and
A231: u1 `2 <= (G * (i1,i2)) `2 ;
A232: 2 in dom g2 by A227, FINSEQ_3:27;
then g2 /. 2 in rng g2 by PARTFUN2:4;
then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153;
then consider u2 being Point of (TOP-REAL 2) such that
A233: g2 /. 2 = u2 and
A234: u2 `1 = (G * (i1,i2)) `1 and
(G * (i1,j2)) `2 <= u2 `2 and
A235: u2 `2 <= (G * (i1,i2)) `2 ;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } ;
u2 `2 < u1 `2 by A107, A228, A232, A229, A233;
then A236: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A230, A231;
u2 = |[(u2 `1),(u2 `2)]| by EUCLID:57;
then A237: LSeg ((G * (i1,i2)),(g2 /. 2)) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A83, A233, A234, A235, TOPREAL3:15;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then g /. n = (f | k) /. (len (f | k)) by A46, A225, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
then A238: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A197, A202, A226, A229, TOPREAL1:def 5;
2 <= len g2 by A198, A203, A225, XREAL_1:8;
then g /. (n + 2) = g2 /. 2 by A225, SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg (u1,u2) by A198, A205, A226, A229, A233, TOPREAL1:def 5;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A226, A229, A233, A236, A238, A237, TOPREAL1:14; :: thesis: verum

end;

suppose len g1 <> n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A239: len g1 < n by A224, XXREAL_0:1;
then (len g1) + 1 <= n by NAT_1:13;
then A240: 1 <= n1 by XREAL_1:21;
n1 + (len g1) = n ;
then A241: LSeg (g,n) = LSeg (g2,n1) by A202, A239, GOBOARD2:10;
A242: n + 1 = (n1 + 1) + (len g1) ;
(n1 + 1) + (len g1) = n + 1 ;
then n1 + 1 <= len g2 by A202, A203, XREAL_1:8;
then A243: g /. (n + 1) = g2 /. (n1 + 1) by A242, NAT_1:11, SEQ_4:153;
len g1 < n + 1 by A201, A239, XXREAL_0:2;
then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A199, A242, GOBOARD2:10;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A180, A204, A241, A243, A240; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

A244: L~ g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g2 or x in LSeg (f,k) )
set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume x in L~ g2 ; :: thesis: x in LSeg (f,k)
then consider X being set such that
A245: x in X and
A246: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A247: X = LSeg (g2,m) and
A248: ( 1 <= m & m + 1 <= len g2 ) by A246;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A249: LSeg (g2,m) = LSeg (q1,q2) by A248, TOPREAL1:def 5;
m + 1 in dom g2 by A248, SEQ_4:151;
then A250: g2 /. (m + 1) in rng g2 by PARTFUN2:4;
m in dom g2 by A248, SEQ_4:151;
then g2 /. m in rng g2 by PARTFUN2:4;
then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (i1,j2))) by A172, A153, A250, TOPREAL1:12;
hence x in LSeg (f,k) by A172, A245, A247, A249; :: thesis: verum

end;

A251: (L~ g1) /\ (L~ g2) = {}

proof

per cases ( k = 1 or k <> 1 ) ;

suppose k = 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

hence (L~ g1) /\ (L~ g2) = {} by A52; :: thesis: verum

end;

suppose k <> 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

then 1 < k by A24, XXREAL_0:1;
then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:9;
then A252: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A244, XBOOLE_1:26;

now

consider x being Element of (L~ g1) /\ (L~ g2);
assume (L~ g1) /\ (L~ g2) <> {} ; :: thesis: contradiction
then ( x in {(f /. k)} & x in L~ g2 ) by A252, TARSKI:def 3, XBOOLE_0:def 4;
hence contradiction by A134, TARSKI:def 1; :: thesis: verum

end;

hence (L~ g1) /\ (L~ g2) = {} ; :: thesis: verum

end;

end;

end;

for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds
LSeg (g,n) misses LSeg (g,m)

proof

A253: 1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A254: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:86 ;
reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ;
set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ;
set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ;
let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) )
assume that
A255: m > n + 1 and
A256: n in dom g and
A257: n + 1 in dom g and
A258: m in dom g and
A259: m + 1 in dom g ; :: thesis: LSeg (g,n) misses LSeg (g,m)
A260: 1 <= n by A256, FINSEQ_3:27;
j2 + 1 <= i2 by A77, NAT_1:13;
then A261: 1 <= l by XREAL_1:21;
then A262: 1 in dom g2 by A89, FINSEQ_3:27;
then A263: ( qq `1 = (G * (i1,i2)) `1 & qq `2 < (G * (i1,i2)) `2 ) by A95;
A264: g /. ((len g1) + 1) = qq by A89, A261, SEQ_4:153;
A265: (G * (i1,j2)) `2 <= qq `2 by A95, A262;
A266: m + 1 <= len g by A259, FINSEQ_3:27;
A267: 1 <= m + 1 by A259, FINSEQ_3:27;
A268: 1 <= n + 1 by A257, FINSEQ_3:27;
A269: n + 1 <= len g by A257, FINSEQ_3:27;
A270: qq = |[(qq `1),(qq `2)]| by EUCLID:57;
A271: 1 <= m by A258, FINSEQ_3:27;
set ql = { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & qq `2 <= z `2 & z `2 <= (G * (i1,i2)) `2 ) } ;
A272: n <= n + 1 by NAT_1:11;
A273: len g = (len g1) + (len g2) by FINSEQ_1:35;
then (len g1) + 1 <= len g by A89, A261, XREAL_1:9;
then A274: LSeg (g,(len g1)) = LSeg (qq,(G * (i1,i2))) by A253, A254, A264, TOPREAL1:def 5
.= { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & qq `2 <= z `2 & z `2 <= (G * (i1,i2)) `2 ) } by A83, A263, A270, TOPREAL3:15 ;
A275: m <= m + 1 by NAT_1:11;
then A276: n + 1 <= m + 1 by A255, XXREAL_0:2;

now

per cases ( m + 1 <= len g1 or len g1 < m + 1 ) ;

suppose A277: m + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then m <= len g1 by A275, XXREAL_0:2;
then A278: m in dom g1 by A271, FINSEQ_3:27;
m + 1 in dom g1 by A267, A277, FINSEQ_3:27;
then A279: LSeg (g,m) = LSeg (g1,m) by A278, TOPREAL3:25;
A280: n + 1 <= len g1 by A276, A277, XXREAL_0:2;
then n <= len g1 by A272, XXREAL_0:2;
then A281: n in dom g1 by A260, FINSEQ_3:27;
n + 1 in dom g1 by A268, A280, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A281, TOPREAL3:25;
hence LSeg (g,n) misses LSeg (g,m) by A42, A255, A279, TOPREAL1:def 9; :: thesis: verum

end;

suppose len g1 < m + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A282: len g1 <= m by NAT_1:13;
then reconsider m1 = m - (len g1) as Element of NAT by INT_1:18;

now

per cases ( m = len g1 or m <> len g1 ) ;

suppose A283: m = len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

A284: LSeg (g,m) c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) )
assume x in LSeg (g,m) ; :: thesis: x in LSeg (f,k)
then consider px being Point of (TOP-REAL 2) such that
A285: ( px = x & px `1 = (G * (i1,i2)) `1 ) and
A286: qq `2 <= px `2 and
A287: px `2 <= (G * (i1,i2)) `2 by A274, A283;
(G * (i1,j2)) `2 <= px `2 by A265, A286, XXREAL_0:2;
hence x in LSeg (f,k) by A152, A285, A287; :: thesis: verum

end;

n <= len g1 by A255, A272, A283, XXREAL_0:2;
then A288: n in dom g1 by A260, FINSEQ_3:27;

now

1 < len g1 by A255, A268, A283, XXREAL_0:2;
then A289: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A289, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A290: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A291: n + 1 in dom g1 by A255, A268, A283, FINSEQ_3:27;
then A292: LSeg (g,n) = LSeg (g1,n) by A288, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A255, A260, A283;
then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:92;
then A293: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A290, A284, XBOOLE_1:27;

now

consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
assume A294: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A295: x in LSeg (g,n) by XBOOLE_0:def 4;
x in {(f /. k)} by A293, A294, TARSKI:def 3;
then A296: x = f /. k by TARSKI:def 1;
f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:86;
hence contradiction by A40, A41, A42, A255, A283, A288, A291, A292, A295, A296, GOBOARD2:7; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose m <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A297: len g1 < m by A282, XXREAL_0:1;
then (len g1) + 1 <= m by NAT_1:13;
then A298: 1 <= m1 by XREAL_1:21;
m + 1 = (m1 + 1) + (len g1) ;
then A299: m1 + 1 <= len g2 by A266, A273, XREAL_1:8;
m = m1 + (len g1) ;
then A300: LSeg (g,m) = LSeg (g2,m1) by A266, A297, GOBOARD2:10;
then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A298, A299;
then A301: LSeg (g,m) c= L~ g2 by ZFMISC_1:92;

now

per cases ( n + 1 <= len g1 or len g1 < n + 1 ) ;

suppose A302: n + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then n <= len g1 by A272, XXREAL_0:2;
then A303: n in dom g1 by A260, FINSEQ_3:27;
n + 1 in dom g1 by A268, A302, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A303, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A260, A302;
then LSeg (g,n) c= L~ g1 by ZFMISC_1:92;
then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A251, A301, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose len g1 < n + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A304: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;
A305: (n - (len g1)) + 1 = (n + 1) - (len g1) ;
A306: n = n1 + (len g1) ;

now

per cases ( len g1 = n or n <> len g1 ) ;

suppose A307: len g1 = n ; :: thesis: LSeg (g,n) misses LSeg (g,m)

now

reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ;
consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
set q1l = { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q2 `2 <= v `2 & v `2 <= q1 `2 ) } ;
A308: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
assume A309: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A310: x in LSeg (g,m) by XBOOLE_0:def 4;
x in LSeg (g,n) by A309, XBOOLE_0:def 4;
then A311: ex qx being Point of (TOP-REAL 2) st
( qx = x & qx `1 = (G * (i1,i2)) `1 & qq `2 <= qx `2 & qx `2 <= (G * (i1,i2)) `2 ) by A274, A307;
A312: m1 in dom g2 by A298, A299, SEQ_4:151;
then A313: q1 `1 = (G * (i1,i2)) `1 by A95;
A314: m1 + 1 in dom g2 by A298, A299, SEQ_4:151;
then A315: q2 `1 = (G * (i1,i2)) `1 by A95;
m1 < m1 + 1 by NAT_1:13;
then A316: q2 `2 < q1 `2 by A107, A312, A314;
LSeg (g2,m1) = LSeg (q2,q1) by A298, A299, TOPREAL1:def 5
.= { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q2 `2 <= v `2 & v `2 <= q1 `2 ) } by A313, A315, A316, A308, TOPREAL3:15 ;
then A317: ex qy being Point of (TOP-REAL 2) st
( qy = x & qy `1 = (G * (i1,i2)) `1 & q2 `2 <= qy `2 & qy `2 <= q1 `2 ) by A300, A310;
( m1 > n1 + 1 & n1 + 1 >= 1 ) by A255, A305, NAT_1:11, XREAL_1:11;
then m1 > 1 by XXREAL_0:2;
then q1 `2 < qq `2 by A107, A262, A312;
hence contradiction by A311, A317, XXREAL_0:2; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose n <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then len g1 < n by A304, XXREAL_0:1;
then A318: LSeg (g,n) = LSeg (g2,n1) by A269, A306, GOBOARD2:10;
m1 > n1 + 1 by A255, A305, XREAL_1:11;
hence LSeg (g,n) misses LSeg (g,m) by A133, A300, A318, TOPREAL1:def 9; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

hence g is s.n.c. by GOBOARD2:6; :: thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

now

set p = g1 /. (len g1);
set q = g2 /. 1;
j2 + 1 <= i2 by A77, NAT_1:13;
then 1 <= l by XREAL_1:21;
then 1 in dom g2 by A90, FINSEQ_1:3;
then (g2 /. 1) `1 = (G * (i1,i2)) `1 by A95;
hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:86; :: thesis: verum

end;

hence g is special by A43, A104, GOBOARD2:13; :: thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
thus L~ g = L~ f :: thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ;
set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ;
A319: len g = (len g1) + (len g2) by FINSEQ_1:35;

A320: now

let j be Element of NAT ; :: thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds
( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) )
assume that
A321: len g1 <= j and
A322: j <= len g ; :: thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds
( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 )
reconsider w = j - (len g1) as Element of NAT by A321, INT_1:18;
let p be Point of (TOP-REAL 2); :: thesis: ( p = g /. j implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) )
assume A323: p = g /. j ; :: thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 )

now

per cases ( j = len g1 or j <> len g1 ) ;

suppose A324: j = len g1 ; :: thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 )

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A325: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence p `1 = (G * (i1,i2)) `1 by A323, A324; :: thesis: ( (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 )
thus ( (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 ) by A68, A74, A71, A30, A72, A77, A79, A81, A323, A324, A325, SEQM_3:def 1; :: thesis: p in rng l1
dom l1 = Seg (len l1) by FINSEQ_1:def 3;
hence p in rng l1 by A68, A72, A78, A323, A324, A325, PARTFUN2:4; :: thesis: verum

end;

suppose j <> len g1 ; :: thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 )

then len g1 < j by A321, XXREAL_0:1;
then (len g1) + 1 <= j by NAT_1:13;
then A326: 1 <= w by XREAL_1:21;
A327: w <= len g2 by A319, A322, XREAL_1:22;
then A328: w in dom g2 by A326, FINSEQ_3:27;
w + (len g1) = j ;
then g /. j = g2 /. w by A326, A327, SEQ_4:153;
hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) by A95, A323, A328; :: thesis: verum

end;

end;

end;

hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) ; :: thesis: verum

end;

thus L~ g c= L~ f :: according to XBOOLE_0:def 10 :: thesis: L~ f c= L~ g

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g or x in L~ f )
assume x in L~ g ; :: thesis: x in L~ f
then consider X being set such that
A329: x in X and
A330: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def 4;
consider i being Element of NAT such that
A331: X = LSeg (g,i) and
A332: 1 <= i and
A333: i + 1 <= len g by A330;

per cases ( i + 1 <= len g1 or i + 1 > len g1 ) ;

suppose A334: i + 1 <= len g1 ; :: thesis: x in L~ f

i <= i + 1 by NAT_1:11;
then i <= len g1 by A334, XXREAL_0:2;
then A335: i in dom g1 by A332, FINSEQ_3:27;
1 <= i + 1 by NAT_1:11;
then i + 1 in dom g1 by A334, FINSEQ_3:27;
then X = LSeg (g1,i) by A331, A335, TOPREAL3:25;
then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A332, A334;
then A336: x in L~ (f | k) by A44, A329, TARSKI:def 4;
L~ (f | k) c= L~ f by TOPREAL3:27;
hence x in L~ f by A336; :: thesis: verum

end;

suppose A337: i + 1 > len g1 ; :: thesis: x in L~ f

reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ;
A338: i <= len g by A333, NAT_1:13;
A339: len g1 <= i by A337, NAT_1:13;
then A340: q1 `1 = (G * (i1,i2)) `1 by A320, A338;
A341: q1 `2 <= (G * (i1,i2)) `2 by A320, A339, A338;
A342: (G * (i1,j2)) `2 <= q1 `2 by A320, A339, A338;
q2 `1 = (G * (i1,i2)) `1 by A320, A333, A337;
then A343: q2 = |[(q1 `1),(q2 `2)]| by A340, EUCLID:57;
A344: q2 `2 <= (G * (i1,i2)) `2 by A320, A333, A337;
A345: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A332, A333, EUCLID:57, TOPREAL1:def 5;
A346: (G * (i1,j2)) `2 <= q2 `2 by A320, A333, A337;

now

per cases ( q1 `2 > q2 `2 or q1 `2 = q2 `2 or q1 `2 < q2 `2 ) by XXREAL_0:1;

suppose q1 `2 > q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = q1 `1 & q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) } by A343, A345, TOPREAL3:15;
then consider p2 being Point of (TOP-REAL 2) such that
A347: ( p2 = x & p2 `1 = q1 `1 ) and
A348: ( q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) by A329, A331;
( (G * (i1,j2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,i2)) `2 ) by A341, A346, A348, XXREAL_0:2;
then A349: x in LSeg (f,k) by A152, A340, A347;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A349, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `2 = q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = {q1} by A343, A345, RLTOPSP1:71;
then x = q1 by A329, A331, TARSKI:def 1;
then A350: x in LSeg (f,k) by A152, A340, A342, A341;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A350, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `2 < q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = q1 `1 & q1 `2 <= p1 `2 & p1 `2 <= q2 `2 ) } by A343, A345, TOPREAL3:15;
then consider p2 being Point of (TOP-REAL 2) such that
A351: ( p2 = x & p2 `1 = q1 `1 ) and
A352: ( q1 `2 <= p2 `2 & p2 `2 <= q2 `2 ) by A329, A331;
( (G * (i1,j2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,i2)) `2 ) by A342, A344, A352, XXREAL_0:2;
then A353: x in LSeg (f,k) by A152, A340, A351;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A353, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ f ; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ f or x in L~ g )
assume x in L~ f ; :: thesis: x in L~ g
then A354: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:8;

per cases ( x in L~ (f | k) or x in LSeg (f,k) ) by A354, XBOOLE_0:def 3;

suppose A355: x in L~ (f | k) ; :: thesis: x in L~ g

L~ g1 c= L~ g by GOBOARD2:11;
hence x in L~ g by A44, A355; :: thesis: verum

end;

suppose x in LSeg (f,k) ; :: thesis: x in L~ g

then consider p1 being Point of (TOP-REAL 2) such that
A356: p1 = x and
A357: p1 `1 = (G * (i1,i2)) `1 and
A358: (G * (i1,j2)) `2 <= p1 `2 and
A359: p1 `2 <= (G * (i1,i2)) `2 by A152;
defpred S₃[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds
q `2 >= p1 `2 ) );

A360: now

reconsider n = len g1 as Nat ;
take n = n; :: thesis: S₃[n]
thus S₃[n] :: thesis: verum

proof

thus ( len g1 <= n & n <= len g ) by A319, XREAL_1:33; :: thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds
q `2 >= p1 `2
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A361: len g1 in dom g1 by FINSEQ_3:27;
let q be Point of (TOP-REAL 2); :: thesis: ( q = g /. n implies q `2 >= p1 `2 )
assume q = g /. n ; :: thesis: q `2 >= p1 `2
then q = (f | k) /. (len (f | k)) by A46, A361, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence q `2 >= p1 `2 by A359; :: thesis: verum

end;

end;

A362: for n being Nat st S₃[n] holds
n <= len g ;
consider ma being Nat such that
A363: ( S₃[ma] & ( for n being Nat st S₃[n] holds
n <= ma ) ) from NAT_1:sch 6(A362, A360);
reconsider ma = ma as Element of NAT by ORDINAL1:def 13;

now

per cases ( ma = len g or ma <> len g ) ;

suppose A364: ma = len g ; :: thesis: x in L~ g

j2 + 1 <= i2 by A77, NAT_1:13;
then A365: 1 <= l by XREAL_1:21;
then (len g1) + 1 <= ma by A89, A319, A364, XREAL_1:9;
then A366: len g1 <= ma - 1 by XREAL_1:21;
then 0 + 1 <= ma by XREAL_1:21;
then reconsider m1 = ma - 1 as Element of NAT by INT_1:18;
reconsider q = g /. m1 as Point of (TOP-REAL 2) ;
A367: ma - 1 <= len g by A364, XREAL_1:45;
then A368: q `1 = (G * (i1,i2)) `1 by A320, A366;
A369: (G * (i1,j2)) `2 <= q `2 by A320, A367, A366;
set lq = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= e `2 & e `2 <= q `2 ) } ;
A370: i2 - l = j2 ;
A371: l in dom g2 by A89, A365, FINSEQ_3:27;
then A372: g /. ma = g2 /. l by A89, A319, A364, FINSEQ_4:84
.= G * (i1,j2) by A89, A90, A371, A370 ;
then p1 `2 <= (G * (i1,j2)) `2 by A363;
then A373: p1 `2 = (G * (i1,j2)) `2 by A358, XXREAL_0:1;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A374: 1 <= m1 by A366, XXREAL_0:2;
A375: m1 + 1 = ma ;
then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg ((G * (i1,j2)),q) ) by A364, A372, A374, EUCLID:57, TOPREAL1:def 5;
then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= e `2 & e `2 <= q `2 ) } by A145, A151, A368, A369, TOPREAL3:15;
then A376: p1 in LSeg (g,m1) by A357, A373, A369;
LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A364, A374, A375;
hence x in L~ g by A356, A376, TARSKI:def 4; :: thesis: verum

end;

suppose ma <> len g ; :: thesis: x in L~ g

then ma < len g by A363, XXREAL_0:1;
then A377: ma + 1 <= len g by NAT_1:13;
reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ;
set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa1 `2 <= p2 `2 & p2 `2 <= qa `2 ) } ;
A378: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:57;
A379: p1 `2 <= qa `2 by A363;
A380: len g1 <= ma + 1 by A363, NAT_1:13;
then A381: qa1 `1 = (G * (i1,i2)) `1 by A320, A377;

A382: now

assume p1 `2 <= qa1 `2 ; :: thesis: contradiction
then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds
p1 `2 <= q `2 ;
then ma + 1 <= ma by A363, A377, A380;
hence contradiction by XREAL_1:31; :: thesis: verum

end;

A383: ( qa `1 = (G * (i1,i2)) `1 & qa = |[(qa `1),(qa `2)]| ) by A320, A363, EUCLID:57;
A384: 1 <= ma by A24, A14, A47, A363, NAT_1:13;
then LSeg (g,ma) = LSeg (qa1,qa) by A377, TOPREAL1:def 5
.= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa1 `2 <= p2 `2 & p2 `2 <= qa `2 ) } by A379, A382, A381, A383, A378, TOPREAL3:15, XXREAL_0:2 ;
then A385: x in LSeg (g,ma) by A356, A357, A379, A382;
LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A377, A384;
hence x in L~ g by A385, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

end;

end;

A386: len g = (len g1) + (len g2) by FINSEQ_1:35;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then 1 in dom g1 by FINSEQ_3:27;
hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:83
.= f /. 1 by A27, A25, FINSEQ_4:86 ;
:: thesis: ( g /. (len g) = f /. (len f) & len f <= len g )
j2 + 1 <= i2 by A77, NAT_1:13;
then A387: 1 <= l by XREAL_1:21;
then A388: l in dom g2 by A90, FINSEQ_1:3;
hence g /. (len g) = g2 /. l by A89, A386, FINSEQ_4:84
.= G * (i1,m1) by A89, A90, A388
.= f /. (len f) by A3, A21, A76 ;
:: thesis: len f <= len g
thus len f <= len g by A3, A14, A47, A89, A387, A386, XREAL_1:9; :: thesis: verum

end;

case A389: i2 = j2 ; :: thesis: contradiction

k <> k + 1 ;
hence contradiction by A5, A27, A29, A19, A21, A76, A389, PARTFUN2:17; :: thesis: verum

end;

case A390: i2 < j2 ; :: thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

l1 /. i2 = l1 . i2 by A68, A73, PARTFUN1:def 8;
then A391: l1 /. i2 = G * (i1,i2) by A68, MATRIX_1:def 8;
then A392: (Y_axis l1) . i2 = (G * (i1,i2)) `2 by A68, A30, A72, GOBOARD1:def 4;
l1 /. j2 = l1 . j2 by A74, A73, PARTFUN1:def 8;
then A393: l1 /. j2 = G * (i1,j2) by A74, MATRIX_1:def 8;
then A394: (Y_axis l1) . j2 = (G * (i1,j2)) `2 by A74, A30, A72, GOBOARD1:def 4;
then A395: (G * (i1,i2)) `2 < (G * (i1,j2)) `2 by A68, A74, A71, A30, A72, A390, A392, SEQM_3:def 1;
reconsider l = j2 - i2 as Element of NAT by A390, INT_1:18;
deffunc H₁( Nat) -> Element of the U1 of (TOP-REAL 2) = G * (i1,(i2 + $1));
consider g2 being FinSequence of (TOP-REAL 2) such that
A396: ( len g2 = l & ( for n being Nat st n in dom g2 holds
g2 /. n = H₁(n) ) ) from FINSEQ_4:sch 2();
take g = g1 ^ g2; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A397: dom g2 = Seg (len g2) by FINSEQ_1:def 3;

A398: now

let n be Element of NAT ; :: thesis: ( n in Seg l implies ( i2 + n in Seg (width G) & [i1,(i2 + n)] in Indices G ) )
A399: n <= i2 + n by NAT_1:11;
assume A400: n in Seg l ; :: thesis: ( i2 + n in Seg (width G) & [i1,(i2 + n)] in Indices G )
then n <= l by FINSEQ_1:3;
then A401: i2 + n <= l + i2 by XREAL_1:9;
j2 <= width G by A74, FINSEQ_1:3;
then A402: i2 + n <= width G by A401, XXREAL_0:2;
1 <= n by A400, FINSEQ_1:3;
then 1 <= i2 + n by A399, XXREAL_0:2;
hence i2 + n in Seg (width G) by A402, FINSEQ_1:3; :: thesis: [i1,(i2 + n)] in Indices G
hence [i1,(i2 + n)] in Indices G by A22, A66, ZFMISC_1:106; :: thesis: verum

end;

now

let n be Element of NAT ; :: thesis: ( n in dom g2 implies ex m being Element of NAT ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) ) )
assume A403: n in dom g2 ; :: thesis: ex m being Element of NAT ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
take m = i1; :: thesis: ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
take k = i2 + n; :: thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) )
thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A396, A398, A397, A403; :: thesis: verum

end;

then A404: for n being Element of NAT st n in dom g holds
ex i, j being Element of NAT st
( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:39;
A405: (X_axis l1) . i2 = (G * (i1,i2)) `1 by A68, A62, A72, A391, GOBOARD1:def 3;

A406: now

let n be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds
( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 )
let p be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & g2 /. n = p implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 ) )
assume that
A407: n in dom g2 and
A408: g2 /. n = p ; :: thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 )
A409: g2 /. n = G * (i1,(i2 + n)) by A396, A407;
set n1 = i2 + n;
set pn = G * (i1,(i2 + n));
A410: i2 + n in Seg (width G) by A396, A398, A397, A407;
then A411: (X_axis l1) . (i2 + n) = (X_axis l1) . i2 by A68, A67, A62, A72, SEQM_3:def 15;
l1 /. (i2 + n) = l1 . (i2 + n) by A73, A396, A398, A397, A407, PARTFUN1:def 8;
then A412: l1 /. (i2 + n) = G * (i1,(i2 + n)) by A410, MATRIX_1:def 8;
then A413: (Y_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `2 by A30, A72, A410, GOBOARD1:def 4;
n <= len g2 by A397, A407, FINSEQ_1:3;
then A414: i2 + n <= i2 + (len g2) by XREAL_1:9;
(X_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `1 by A62, A72, A410, A412, GOBOARD1:def 3;
hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 ) by A68, A74, A71, A30, A72, A396, A392, A394, A405, A408, A409, A410, A414, A411, A413, SEQ_4:154, XREAL_1:33; :: thesis: ( p in rng l1 & p `2 > (G * (i1,i2)) `2 )
dom l1 = Seg (len l1) by FINSEQ_1:def 3;
hence p in rng l1 by A72, A408, A409, A410, A412, PARTFUN2:4; :: thesis: p `2 > (G * (i1,i2)) `2
1 <= n by A397, A407, FINSEQ_1:3;
then i2 < i2 + n by XREAL_1:31;
hence p `2 > (G * (i1,i2)) `2 by A68, A71, A30, A72, A392, A408, A409, A410, A413, SEQM_3:def 1; :: thesis: verum

end;

A415: g2 is special

proof

let n be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
set p = g2 /. n;
assume A416: ( 1 <= n & n + 1 <= len g2 ) ; :: thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
then n in dom g2 by SEQ_4:151;
then A417: (g2 /. n) `1 = (G * (i1,i2)) `1 by A406;
n + 1 in dom g2 by A416, SEQ_4:151;
hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A406, A417; :: thesis: verum

end;

now

let n, m be Element of NAT ; :: thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m )
assume that
A418: ( n in dom g2 & m in dom g2 ) and
A419: n <> m ; :: thesis: not g2 /. n = g2 /. m
A420: ( g2 /. n = G * (i1,(i2 + n)) & g2 /. m = G * (i1,(i2 + m)) ) by A396, A418;
assume A421: g2 /. n = g2 /. m ; :: thesis: contradiction
( [i1,(i2 + n)] in Indices G & [i1,(i2 + m)] in Indices G ) by A396, A398, A397, A418;
then i2 + n = i2 + m by A420, A421, GOBOARD1:21;
hence contradiction by A419; :: thesis: verum

end;

then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds
n = m ;
then A422: g2 is one-to-one by PARTFUN2:16;
set lk = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } ;
A423: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:57;

A424: now

let n, m be Element of NAT ; :: thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds
p `2 < q `2
let p, q be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies p `2 < q `2 )
assume that
A425: n in dom g2 and
A426: m in dom g2 and
A427: n < m and
A428: ( g2 /. n = p & g2 /. m = q ) ; :: thesis: p `2 < q `2
A429: i2 + n in Seg (width G) by A396, A398, A397, A425;
set n1 = i2 + n;
set m1 = i2 + m;
set pn = G * (i1,(i2 + n));
set pm = G * (i1,(i2 + m));
A430: i2 + n < i2 + m by A427, XREAL_1:10;
l1 /. (i2 + n) = l1 . (i2 + n) by A73, A396, A398, A397, A425, PARTFUN1:def 8;
then l1 /. (i2 + n) = G * (i1,(i2 + n)) by A429, MATRIX_1:def 8;
then A431: (Y_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `2 by A30, A72, A429, GOBOARD1:def 4;
A432: i2 + m in Seg (width G) by A396, A398, A397, A426;
l1 /. (i2 + m) = l1 . (i2 + m) by A73, A396, A398, A397, A426, PARTFUN1:def 8;
then l1 /. (i2 + m) = G * (i1,(i2 + m)) by A432, MATRIX_1:def 8;
then A433: (Y_axis l1) . (i2 + m) = (G * (i1,(i2 + m))) `2 by A30, A72, A432, GOBOARD1:def 4;
( g2 /. n = G * (i1,(i2 + n)) & g2 /. m = G * (i1,(i2 + m)) ) by A396, A425, A426;
hence p `2 < q `2 by A71, A30, A72, A428, A429, A432, A430, A431, A433, SEQM_3:def 1; :: thesis: verum

end;

for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds
LSeg (g2,n) misses LSeg (g2,m)

proof

let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) )
assume that
A434: m > n + 1 and
A435: n in dom g2 and
A436: n + 1 in dom g2 and
A437: m in dom g2 and
A438: m + 1 in dom g2 and
A439: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A440: ( p1 `1 = (G * (i1,i2)) `1 & p2 `1 = (G * (i1,i2)) `1 ) by A406, A435, A436;
n < n + 1 by NAT_1:13;
then A441: p1 `2 < p2 `2 by A424, A435, A436;
set lp = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p1 `2 <= w `2 & w `2 <= p2 `2 ) } ;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } ;
A442: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:57;
m < m + 1 by NAT_1:13;
then A443: q1 `2 < q2 `2 by A424, A437, A438;
A444: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
consider x being Element of (LSeg (g2,n)) /\ (LSeg (g2,m));
A445: x in LSeg (g2,n) by A439, XBOOLE_0:def 4;
A446: ( q1 `1 = (G * (i1,i2)) `1 & q2 `1 = (G * (i1,i2)) `1 ) by A406, A437, A438;
A447: x in LSeg (g2,m) by A439, XBOOLE_0:def 4;
( 1 <= m & m + 1 <= len g2 ) by A437, A438, FINSEQ_3:27;
then LSeg (g2,m) = LSeg (q1,q2) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } by A443, A446, A444, TOPREAL3:15 ;
then A448: ex tm being Point of (TOP-REAL 2) st
( tm = x & tm `1 = (G * (i1,i2)) `1 & q1 `2 <= tm `2 & tm `2 <= q2 `2 ) by A447;
( 1 <= n & n + 1 <= len g2 ) by A435, A436, FINSEQ_3:27;
then LSeg (g2,n) = LSeg (p1,p2) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p1 `2 <= w `2 & w `2 <= p2 `2 ) } by A441, A440, A442, TOPREAL3:15 ;
then A449: ex tn being Point of (TOP-REAL 2) st
( tn = x & tn `1 = (G * (i1,i2)) `1 & p1 `2 <= tn `2 & tn `2 <= p2 `2 ) by A445;
p2 `2 < q1 `2 by A424, A434, A436, A437;
hence contradiction by A449, A448, XXREAL_0:2; :: thesis: verum

end;

then A450: g2 is s.n.c. by GOBOARD2:6;
A451: not f /. k in L~ g2

proof

set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume f /. k in L~ g2 ; :: thesis: contradiction
then consider X being set such that
A452: f /. k in X and
A453: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A454: X = LSeg (g2,m) and
A455: ( 1 <= m & m + 1 <= len g2 ) by A453;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A456: m in dom g2 by A455, SEQ_4:151;
then A457: q1 `1 = (G * (i1,i2)) `1 by A406;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } ;
A458: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
A459: m + 1 in dom g2 by A455, SEQ_4:151;
then A460: q2 `1 = (G * (i1,i2)) `1 by A406;
m < m + 1 by NAT_1:13;
then A461: q1 `2 < q2 `2 by A424, A456, A459;
LSeg (g2,m) = LSeg (q1,q2) by A455, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } by A457, A460, A461, A458, TOPREAL3:15 ;
then ex p being Point of (TOP-REAL 2) st
( p = f /. k & p `1 = (G * (i1,i2)) `1 & q1 `2 <= p `2 & p `2 <= q2 `2 ) by A452, A454;
hence contradiction by A29, A406, A456; :: thesis: verum

end;

(X_axis l1) . j2 = (G * (i1,j2)) `1 by A74, A62, A72, A393, GOBOARD1:def 3;
then A462: (G * (i1,i2)) `1 = (G * (i1,j2)) `1 by A68, A74, A67, A62, A72, A405, SEQM_3:def 15;

A463: now

let n be Element of NAT ; :: thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A464: n in dom g2 and
A465: n + 1 in dom g2 ; :: thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1
let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A466: [l1,l2] in Indices G and
A467: [l3,l4] in Indices G and
A468: g2 /. n = G * (l1,l2) and
A469: g2 /. (n + 1) = G * (l3,l4) ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( g2 /. (n + 1) = G * (i1,(i2 + (n + 1))) & [i1,(i2 + (n + 1))] in Indices G ) by A396, A398, A397, A465;
then A470: ( l3 = i1 & l4 = i2 + (n + 1) ) by A467, A469, GOBOARD1:21;
( g2 /. n = G * (i1,(i2 + n)) & [i1,(i2 + n)] in Indices G ) by A396, A398, A397, A464;
then ( l1 = i1 & l2 = i2 + n ) by A466, A468, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs ((i2 + n) - (i2 + (n + 1)))) by A470, ABSVALUE:7
.= abs (- 1)
.= abs 1 by COMPLEX1:138
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

now

let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A471: [l1,l2] in Indices G and
A472: [l3,l4] in Indices G and
A473: g1 /. (len g1) = G * (l1,l2) and
A474: g2 /. 1 = G * (l3,l4) and
len g1 in dom g1 and
A475: 1 in dom g2 ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( g2 /. 1 = G * (i1,(i2 + 1)) & [i1,(i2 + 1)] in Indices G ) by A396, A398, A397, A475;
then A476: ( l3 = i1 & l4 = i2 + 1 ) by A472, A474, GOBOARD1:21;
(f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:86;
then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A471, A473, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs (i2 - (i2 + 1))) by A476, ABSVALUE:7
.= abs ((i2 - i2) + (- 1))
.= abs 1 by COMPLEX1:138
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

then for n being Element of NAT st n in dom g & n + 1 in dom g holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A48, A463, GOBOARD1:40;
hence g is_sequence_on G by A404, GOBOARD1:def 11; :: thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A477: G * (i1,j2) = |[((G * (i1,j2)) `1),((G * (i1,j2)) `2)]| by EUCLID:57;
A478: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A395, A462, A423, A477, TOPREAL3:15 ;
A479: rng g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g2 or x in LSeg (f,k) )
assume x in rng g2 ; :: thesis: x in LSeg (f,k)
then consider n being Element of NAT such that
A480: n in dom g2 and
A481: g2 /. n = x by PARTFUN2:4;
set pn = G * (i1,(i2 + n));
A482: g2 /. n = G * (i1,(i2 + n)) by A396, A480;
then A483: (G * (i1,(i2 + n))) `2 <= (G * (i1,j2)) `2 by A406, A480;
( (G * (i1,(i2 + n))) `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= (G * (i1,(i2 + n))) `2 ) by A406, A480, A482;
hence x in LSeg (f,k) by A478, A481, A482, A483; :: thesis: verum

end;

A484: not f /. k in rng g2

proof

assume f /. k in rng g2 ; :: thesis: contradiction
then consider n being Element of NAT such that
A485: n in dom g2 and
A486: g2 /. n = f /. k by PARTFUN2:4;
A487: 0 < n by A485, FINSEQ_3:27;
A488: g2 /. n = G * (i1,(i2 + n)) by A396, A485;
dom g2 = Seg (len g2) by FINSEQ_1:def 3;
then [i1,(i2 + n)] in Indices G by A396, A398, A485;
then i2 + n = i2 by A28, A29, A486, A488, GOBOARD1:21;
hence contradiction by A487; :: thesis: verum

end;

(rng g1) /\ (rng g2) = {}

proof

consider x being Element of (rng g1) /\ (rng g2);
assume A489: not (rng g1) /\ (rng g2) = {} ; :: thesis: contradiction
then A490: x in rng g2 by XBOOLE_0:def 4;
A491: x in rng g1 by A489, XBOOLE_0:def 4;

now

per cases ( k = 1 or 1 < k ) by A24, XXREAL_0:1;

suppose k = 1 ; :: thesis: contradiction

hence contradiction by A52, A484, A491, A490, TARSKI:def 1; :: thesis: verum

end;

suppose 1 < k ; :: thesis: contradiction

then ( x in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A479, A491, A490, GOBOARD2:9, XBOOLE_0:def 4;
hence contradiction by A484, A490, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

then rng g1 misses rng g2 by XBOOLE_0:def 7;
hence g is one-to-one by A40, A422, FINSEQ_3:98; :: thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A492: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def 5;
A493: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds
(LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}

proof

let n be Element of NAT ; :: thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} )
assume that
A494: 1 <= n and
A495: n + 2 <= len g2 ; :: thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}
A496: n + 1 in dom g2 by A494, A495, SEQ_4:152;
then g2 /. (n + 1) in rng g2 by PARTFUN2:4;
then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u1 being Point of (TOP-REAL 2) such that
A497: g2 /. (n + 1) = u1 and
A498: u1 `1 = (G * (i1,i2)) `1 and
(G * (i1,i2)) `2 <= u1 `2 and
u1 `2 <= (G * (i1,j2)) `2 ;
A499: n + 2 in dom g2 by A494, A495, SEQ_4:152;
then g2 /. (n + 2) in rng g2 by PARTFUN2:4;
then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u2 being Point of (TOP-REAL 2) such that
A500: g2 /. (n + 2) = u2 and
A501: u2 `1 = (G * (i1,i2)) `1 and
(G * (i1,i2)) `2 <= u2 `2 and
u2 `2 <= (G * (i1,j2)) `2 ;
( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11;
then A502: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A495, A497, A500, TOPREAL1:def 5;
n + 1 < (n + 1) + 1 by NAT_1:13;
then A503: u1 `2 < u2 `2 by A424, A496, A499, A497, A500;
A504: n in dom g2 by A494, A495, SEQ_4:152;
then g2 /. n in rng g2 by PARTFUN2:4;
then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u being Point of (TOP-REAL 2) such that
A505: g2 /. n = u and
A506: u `1 = (G * (i1,i2)) `1 and
(G * (i1,i2)) `2 <= u `2 and
u `2 <= (G * (i1,j2)) `2 ;
n + 1 <= n + 2 by XREAL_1:8;
then n + 1 <= len g2 by A495, XXREAL_0:2;
then A507: LSeg (g2,n) = LSeg (u,u1) by A494, A505, A497, TOPREAL1:def 5;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } ;
n < n + 1 by NAT_1:13;
then A508: u `2 < u1 `2 by A424, A504, A496, A505, A497;
then A509: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } by A498, A503;
( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:57;
then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } by A505, A506, A500, A501, A503, A508, TOPREAL3:15, XXREAL_0:2;
hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A505, A497, A500, A507, A502, A509, TOPREAL1:14; :: thesis: verum

end;

thus g is unfolded :: thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

let n be Nat; :: according to TOPREAL1:def 8 :: thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} )
assume that
A510: 1 <= n and
A511: n + 2 <= len g ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}
A512: (n + 1) + 1 <= len g by A511;
n + 1 <= (n + 1) + 1 by NAT_1:11;
then A513: n + 1 <= len g by A511, XXREAL_0:2;
A514: len g = (len g1) + (len g2) by FINSEQ_1:35;
(n + 2) - (len g1) = (n - (len g1)) + 2 ;
then A515: (n - (len g1)) + 2 <= len g2 by A511, A514, XREAL_1:22;
A516: 1 <= n + 1 by NAT_1:11;
A517: n <= n + 1 by NAT_1:11;
A518: n + (1 + 1) = (n + 1) + 1 ;

per cases ( n + 2 <= len g1 or len g1 < n + 2 ) ;

suppose A519: n + 2 <= len g1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

A520: n + (1 + 1) = (n + 1) + 1 ;
A521: n + 1 in dom g1 by A510, A519, SEQ_4:152;
then A522: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:83;
n in dom g1 by A510, A519, SEQ_4:152;
then A523: LSeg (g1,n) = LSeg (g,n) by A521, TOPREAL3:25;
n + 2 in dom g1 by A510, A519, SEQ_4:152;
then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A521, A520, TOPREAL3:25;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A510, A519, A523, A522, TOPREAL1:def 8; :: thesis: verum

end;

suppose len g1 < n + 2 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then (len g1) + 1 <= n + 2 by NAT_1:13;
then A524: len g1 <= (n + 2) - 1 by XREAL_1:21;

now

per cases ( len g1 = n + 1 or len g1 <> n + 1 ) ;

suppose A525: len g1 = n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

now

1 < len g1 by A510, A525, NAT_1:13;
then A526: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A526, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A527: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A510, A511, A516, A513, A518, TOPREAL1:27;
then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def 4;
then A528: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:37;
A529: 1 <= (len g) - (len g1) by A512, A525, XREAL_1:21;
then 1 in dom g2 by A514, FINSEQ_3:27;
then A530: g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u1 being Point of (TOP-REAL 2) such that
A531: g2 /. 1 = u1 and
u1 `1 = (G * (i1,i2)) `1 and
(G * (i1,i2)) `2 <= u1 `2 and
u1 `2 <= (G * (i1,j2)) `2 ;
G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (i1,j2))) by RLTOPSP1:69;
then A532: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A492, A479, A530, A531, TOPREAL1:12;
1 <= n + 1 by NAT_1:11;
then A533: n + 1 in dom g1 by A525, FINSEQ_3:27;
then A534: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A525, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
n in dom g1 by A510, A517, A525, FINSEQ_3:27;
then A535: LSeg (g,n) = LSeg (g1,n) by A533, TOPREAL3:25;
g /. (n + 2) = g2 /. 1 by A518, A514, A525, A529, SEQ_4:153;
then A536: LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A511, A516, A518, A534, A531, TOPREAL1:def 5;
LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:26;
then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A527, A535, A534, A532, A536, XBOOLE_1:27;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A528, XBOOLE_0:def 10; :: thesis: verum

end;

suppose len g1 <> n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then len g1 < n + 1 by A524, XXREAL_0:1;
then A537: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;

now

per cases ( len g1 = n or len g1 <> n ) ;

suppose A538: len g1 = n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A539: 2 <= len g2 by A511, A514, XREAL_1:8;
then 1 <= len g2 by XXREAL_0:2;
then A540: g /. (n + 1) = g2 /. 1 by A538, SEQ_4:153;
1 <= len g2 by A539, XXREAL_0:2;
then A541: 1 in dom g2 by FINSEQ_3:27;
then g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u1 being Point of (TOP-REAL 2) such that
A542: g2 /. 1 = u1 and
A543: ( u1 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= u1 `2 ) and
u1 `2 <= (G * (i1,j2)) `2 ;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then g /. n = (f | k) /. (len (f | k)) by A46, A538, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
then A544: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A510, A513, A540, A542, TOPREAL1:def 5;
A545: 2 in dom g2 by A539, FINSEQ_3:27;
then g2 /. 2 in rng g2 by PARTFUN2:4;
then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479;
then consider u2 being Point of (TOP-REAL 2) such that
A546: g2 /. 2 = u2 and
A547: ( u2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= u2 `2 ) and
u2 `2 <= (G * (i1,j2)) `2 ;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= u2 `2 ) } ;
u1 `2 < u2 `2 by A424, A541, A545, A542, A546;
then ( u2 = |[(u2 `1),(u2 `2)]| & u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= u2 `2 ) } ) by A543, EUCLID:57;
then A548: u1 in LSeg ((G * (i1,i2)),u2) by A423, A547, TOPREAL3:15;
g /. (n + 2) = g2 /. 2 by A538, A539, SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg (u1,u2) by A511, A516, A518, A540, A542, A546, TOPREAL1:def 5;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A540, A542, A544, A548, TOPREAL1:14; :: thesis: verum

end;

suppose len g1 <> n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A549: len g1 < n by A537, XXREAL_0:1;
then (len g1) + 1 <= n by NAT_1:13;
then A550: 1 <= n1 by XREAL_1:21;
n1 + (len g1) = n ;
then A551: LSeg (g,n) = LSeg (g2,n1) by A513, A549, GOBOARD2:10;
A552: (n1 + 1) + (len g1) = n + 1 ;
then n1 + 1 <= len g2 by A513, A514, XREAL_1:8;
then A553: g /. (n + 1) = g2 /. (n1 + 1) by A552, NAT_1:11, SEQ_4:153;
len g1 < n + 1 by A517, A549, XXREAL_0:2;
then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A512, A552, GOBOARD2:10;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A493, A515, A551, A553, A550; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

A554: L~ g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g2 or x in LSeg (f,k) )
set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume x in L~ g2 ; :: thesis: x in LSeg (f,k)
then consider X being set such that
A555: x in X and
A556: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A557: X = LSeg (g2,m) and
A558: ( 1 <= m & m + 1 <= len g2 ) by A556;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A559: LSeg (g2,m) = LSeg (q1,q2) by A558, TOPREAL1:def 5;
m + 1 in dom g2 by A558, SEQ_4:151;
then A560: g2 /. (m + 1) in rng g2 by PARTFUN2:4;
m in dom g2 by A558, SEQ_4:151;
then g2 /. m in rng g2 by PARTFUN2:4;
then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (i1,j2))) by A492, A479, A560, TOPREAL1:12;
hence x in LSeg (f,k) by A492, A555, A557, A559; :: thesis: verum

end;

A561: (L~ g1) /\ (L~ g2) = {}

proof

per cases ( k = 1 or k <> 1 ) ;

suppose k = 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

hence (L~ g1) /\ (L~ g2) = {} by A52; :: thesis: verum

end;

suppose k <> 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

then 1 < k by A24, XXREAL_0:1;
then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:9;
then A562: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A554, XBOOLE_1:26;

now

consider x being Element of (L~ g1) /\ (L~ g2);
assume (L~ g1) /\ (L~ g2) <> {} ; :: thesis: contradiction
then ( x in {(f /. k)} & x in L~ g2 ) by A562, TARSKI:def 3, XBOOLE_0:def 4;
hence contradiction by A451, TARSKI:def 1; :: thesis: verum

end;

hence (L~ g1) /\ (L~ g2) = {} ; :: thesis: verum

end;

end;

end;

for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds
LSeg (g,n) misses LSeg (g,m)

proof

A563: 1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A564: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:86 ;
reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ;
set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ;
set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ;
let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) )
assume that
A565: m > n + 1 and
A566: n in dom g and
A567: n + 1 in dom g and
A568: m in dom g and
A569: m + 1 in dom g ; :: thesis: LSeg (g,n) misses LSeg (g,m)
A570: 1 <= n by A566, FINSEQ_3:27;
i2 + 1 <= j2 by A390, NAT_1:13;
then A571: 1 <= l by XREAL_1:21;
then A572: 1 in dom g2 by A396, FINSEQ_3:27;
then A573: ( qq `1 = (G * (i1,i2)) `1 & qq `2 > (G * (i1,i2)) `2 ) by A406;
A574: g /. ((len g1) + 1) = qq by A396, A571, SEQ_4:153;
A575: qq `2 <= (G * (i1,j2)) `2 by A406, A572;
A576: m + 1 <= len g by A569, FINSEQ_3:27;
A577: 1 <= m + 1 by A569, FINSEQ_3:27;
A578: 1 <= n + 1 by A567, FINSEQ_3:27;
A579: n + 1 <= len g by A567, FINSEQ_3:27;
A580: qq = |[(qq `1),(qq `2)]| by EUCLID:57;
A581: 1 <= m by A568, FINSEQ_3:27;
set ql = { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= z `2 & z `2 <= qq `2 ) } ;
A582: n <= n + 1 by NAT_1:11;
A583: len g = (len g1) + (len g2) by FINSEQ_1:35;
then (len g1) + 1 <= len g by A396, A571, XREAL_1:9;
then A584: LSeg (g,(len g1)) = LSeg ((G * (i1,i2)),qq) by A563, A564, A574, TOPREAL1:def 5
.= { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= z `2 & z `2 <= qq `2 ) } by A423, A573, A580, TOPREAL3:15 ;
A585: m <= m + 1 by NAT_1:11;
then A586: n + 1 <= m + 1 by A565, XXREAL_0:2;

now

per cases ( m + 1 <= len g1 or len g1 < m + 1 ) ;

suppose A587: m + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then m <= len g1 by A585, XXREAL_0:2;
then A588: m in dom g1 by A581, FINSEQ_3:27;
m + 1 in dom g1 by A577, A587, FINSEQ_3:27;
then A589: LSeg (g,m) = LSeg (g1,m) by A588, TOPREAL3:25;
A590: n + 1 <= len g1 by A586, A587, XXREAL_0:2;
then n <= len g1 by A582, XXREAL_0:2;
then A591: n in dom g1 by A570, FINSEQ_3:27;
n + 1 in dom g1 by A578, A590, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A591, TOPREAL3:25;
hence LSeg (g,n) misses LSeg (g,m) by A42, A565, A589, TOPREAL1:def 9; :: thesis: verum

end;

suppose len g1 < m + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A592: len g1 <= m by NAT_1:13;
then reconsider m1 = m - (len g1) as Element of NAT by INT_1:18;

now

per cases ( m = len g1 or m <> len g1 ) ;

suppose A593: m = len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

A594: LSeg (g,m) c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) )
assume x in LSeg (g,m) ; :: thesis: x in LSeg (f,k)
then consider px being Point of (TOP-REAL 2) such that
A595: ( px = x & px `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= px `2 ) and
A596: px `2 <= qq `2 by A584, A593;
(G * (i1,j2)) `2 >= px `2 by A575, A596, XXREAL_0:2;
hence x in LSeg (f,k) by A478, A595; :: thesis: verum

end;

n <= len g1 by A565, A582, A593, XXREAL_0:2;
then A597: n in dom g1 by A570, FINSEQ_3:27;

now

1 < len g1 by A565, A578, A593, XXREAL_0:2;
then A598: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A598, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A599: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A600: n + 1 in dom g1 by A565, A578, A593, FINSEQ_3:27;
then A601: LSeg (g,n) = LSeg (g1,n) by A597, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A565, A570, A593;
then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:92;
then A602: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A599, A594, XBOOLE_1:27;

now

consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
assume A603: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A604: x in LSeg (g,n) by XBOOLE_0:def 4;
x in {(f /. k)} by A602, A603, TARSKI:def 3;
then A605: x = f /. k by TARSKI:def 1;
f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:86;
hence contradiction by A40, A41, A42, A565, A593, A597, A600, A601, A604, A605, GOBOARD2:7; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose m <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A606: len g1 < m by A592, XXREAL_0:1;
then (len g1) + 1 <= m by NAT_1:13;
then A607: 1 <= m1 by XREAL_1:21;
m + 1 = (m1 + 1) + (len g1) ;
then A608: m1 + 1 <= len g2 by A576, A583, XREAL_1:8;
m = m1 + (len g1) ;
then A609: LSeg (g,m) = LSeg (g2,m1) by A576, A606, GOBOARD2:10;
then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A607, A608;
then A610: LSeg (g,m) c= L~ g2 by ZFMISC_1:92;

now

per cases ( n + 1 <= len g1 or len g1 < n + 1 ) ;

suppose A611: n + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then n <= len g1 by A582, XXREAL_0:2;
then A612: n in dom g1 by A570, FINSEQ_3:27;
n + 1 in dom g1 by A578, A611, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A612, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A570, A611;
then LSeg (g,n) c= L~ g1 by ZFMISC_1:92;
then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A561, A610, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose len g1 < n + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A613: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;
A614: (n - (len g1)) + 1 = (n + 1) - (len g1) ;
A615: n = n1 + (len g1) ;

now

per cases ( len g1 = n or n <> len g1 ) ;

suppose A616: len g1 = n ; :: thesis: LSeg (g,n) misses LSeg (g,m)

now

reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ;
consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
set q1l = { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q1 `2 <= v `2 & v `2 <= q2 `2 ) } ;
A617: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
assume A618: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A619: x in LSeg (g,m) by XBOOLE_0:def 4;
x in LSeg (g,n) by A618, XBOOLE_0:def 4;
then A620: ex qx being Point of (TOP-REAL 2) st
( qx = x & qx `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= qx `2 & qx `2 <= qq `2 ) by A584, A616;
A621: m1 in dom g2 by A607, A608, SEQ_4:151;
then A622: q1 `1 = (G * (i1,i2)) `1 by A406;
A623: m1 + 1 in dom g2 by A607, A608, SEQ_4:151;
then A624: q2 `1 = (G * (i1,i2)) `1 by A406;
m1 < m1 + 1 by NAT_1:13;
then A625: q1 `2 < q2 `2 by A424, A621, A623;
LSeg (g2,m1) = LSeg (q1,q2) by A607, A608, TOPREAL1:def 5
.= { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q1 `2 <= v `2 & v `2 <= q2 `2 ) } by A622, A624, A625, A617, TOPREAL3:15 ;
then A626: ex qy being Point of (TOP-REAL 2) st
( qy = x & qy `1 = (G * (i1,i2)) `1 & q1 `2 <= qy `2 & qy `2 <= q2 `2 ) by A609, A619;
( m1 > n1 + 1 & n1 + 1 >= 1 ) by A565, A614, NAT_1:11, XREAL_1:11;
then m1 > 1 by XXREAL_0:2;
then qq `2 < q1 `2 by A424, A572, A621;
hence contradiction by A620, A626, XXREAL_0:2; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose n <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then len g1 < n by A613, XXREAL_0:1;
then A627: LSeg (g,n) = LSeg (g2,n1) by A579, A615, GOBOARD2:10;
m1 > n1 + 1 by A565, A614, XREAL_1:11;
hence LSeg (g,n) misses LSeg (g,m) by A450, A609, A627, TOPREAL1:def 9; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

hence g is s.n.c. by GOBOARD2:6; :: thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

now

set p = g1 /. (len g1);
set q = g2 /. 1;
i2 + 1 <= j2 by A390, NAT_1:13;
then 1 <= l by XREAL_1:21;
then 1 in dom g2 by A396, FINSEQ_3:27;
then (g2 /. 1) `1 = (G * (i1,i2)) `1 by A406;
hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:86; :: thesis: verum

end;

hence g is special by A43, A415, GOBOARD2:13; :: thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
thus L~ g = L~ f :: thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ;
set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ;
A628: len g = (len g1) + (len g2) by FINSEQ_1:35;

A629: now

let j be Element of NAT ; :: thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds
( b₃ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>3</sub> `2 & b₃ `2 <= (G * (i1,j2)) `2 & b₃ in rng l1 ) )
assume that
A630: len g1 <= j and
A631: j <= len g ; :: thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds
( b₃ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>3</sub> `2 & b₃ `2 <= (G * (i1,j2)) `2 & b₃ in rng l1 )
reconsider w = j - (len g1) as Element of NAT by A630, INT_1:18;
let p be Point of (TOP-REAL 2); :: thesis: ( p = g /. j implies ( b₂ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>2</sub> `2 & b₂ `2 <= (G * (i1,j2)) `2 & b₂ in rng l1 ) )
assume A632: p = g /. j ; :: thesis: ( b₂ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>2</sub> `2 & b₂ `2 <= (G * (i1,j2)) `2 & b₂ in rng l1 )

per cases ( j = len g1 or j <> len g1 ) ;

suppose A633: j = len g1 ; :: thesis: ( b₂ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>2</sub> `2 & b₂ `2 <= (G * (i1,j2)) `2 & b₂ in rng l1 )

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A634: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence p `1 = (G * (i1,i2)) `1 by A632, A633; :: thesis: ( (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 )
thus ( (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 ) by A68, A74, A71, A30, A72, A390, A392, A394, A632, A633, A634, SEQM_3:def 1; :: thesis: p in rng l1
dom l1 = Seg (len l1) by FINSEQ_1:def 3;
hence p in rng l1 by A68, A72, A391, A632, A633, A634, PARTFUN2:4; :: thesis: verum

end;

suppose j <> len g1 ; :: thesis: ( b₂ `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b<sub>2</sub> `2 & b₂ `2 <= (G * (i1,j2)) `2 & b₂ in rng l1 )

then len g1 < j by A630, XXREAL_0:1;
then (len g1) + 1 <= j by NAT_1:13;
then A635: 1 <= w by XREAL_1:21;
A636: w <= len g2 by A628, A631, XREAL_1:22;
then A637: w in dom g2 by A635, FINSEQ_3:27;
j = w + (len g1) ;
then g /. j = g2 /. w by A635, A636, SEQ_4:153;
hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 ) by A406, A632, A637; :: thesis: verum

end;

end;

end;

thus L~ g c= L~ f :: according to XBOOLE_0:def 10 :: thesis: L~ f c= L~ g

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g or x in L~ f )
assume x in L~ g ; :: thesis: x in L~ f
then consider X being set such that
A638: x in X and
A639: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def 4;
consider i being Element of NAT such that
A640: X = LSeg (g,i) and
A641: 1 <= i and
A642: i + 1 <= len g by A639;

per cases ( i + 1 <= len g1 or i + 1 > len g1 ) ;

suppose A643: i + 1 <= len g1 ; :: thesis: x in L~ f

i <= i + 1 by NAT_1:11;
then i <= len g1 by A643, XXREAL_0:2;
then A644: i in dom g1 by A641, FINSEQ_3:27;
1 <= i + 1 by NAT_1:11;
then i + 1 in dom g1 by A643, FINSEQ_3:27;
then X = LSeg (g1,i) by A640, A644, TOPREAL3:25;
then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A641, A643;
then A645: x in L~ (f | k) by A44, A638, TARSKI:def 4;
L~ (f | k) c= L~ f by TOPREAL3:27;
hence x in L~ f by A645; :: thesis: verum

end;

suppose A646: i + 1 > len g1 ; :: thesis: x in L~ f

reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ;
A647: i <= len g by A642, NAT_1:13;
A648: len g1 <= i by A646, NAT_1:13;
then A649: q1 `1 = (G * (i1,i2)) `1 by A629, A647;
A650: q1 `2 <= (G * (i1,j2)) `2 by A629, A648, A647;
A651: (G * (i1,i2)) `2 <= q1 `2 by A629, A648, A647;
q2 `1 = (G * (i1,i2)) `1 by A629, A642, A646;
then A652: q2 = |[(q1 `1),(q2 `2)]| by A649, EUCLID:57;
A653: q2 `2 <= (G * (i1,j2)) `2 by A629, A642, A646;
A654: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A641, A642, EUCLID:57, TOPREAL1:def 5;
A655: (G * (i1,i2)) `2 <= q2 `2 by A629, A642, A646;

now

per cases ( q1 `2 > q2 `2 or q1 `2 = q2 `2 or q1 `2 < q2 `2 ) by XXREAL_0:1;

suppose q1 `2 > q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = q1 `1 & q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) } by A652, A654, TOPREAL3:15;
then consider p2 being Point of (TOP-REAL 2) such that
A656: ( p2 = x & p2 `1 = q1 `1 ) and
A657: ( q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) by A638, A640;
( (G * (i1,i2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,j2)) `2 ) by A650, A655, A657, XXREAL_0:2;
then A658: x in LSeg (f,k) by A478, A649, A656;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A658, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `2 = q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = {q1} by A652, A654, RLTOPSP1:71;
then x = q1 by A638, A640, TARSKI:def 1;
then A659: x in LSeg (f,k) by A478, A649, A651, A650;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A659, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `2 < q2 `2 ; :: thesis: x in L~ f

then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = q1 `1 & q1 `2 <= p1 `2 & p1 `2 <= q2 `2 ) } by A652, A654, TOPREAL3:15;
then consider p2 being Point of (TOP-REAL 2) such that
A660: ( p2 = x & p2 `1 = q1 `1 ) and
A661: ( q1 `2 <= p2 `2 & p2 `2 <= q2 `2 ) by A638, A640;
( (G * (i1,i2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,j2)) `2 ) by A651, A653, A661, XXREAL_0:2;
then A662: x in LSeg (f,k) by A478, A649, A660;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A662, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ f ; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ f or x in L~ g )
assume x in L~ f ; :: thesis: x in L~ g
then A663: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:8;

now

per cases ( x in L~ (f | k) or x in LSeg (f,k) ) by A663, XBOOLE_0:def 3;

suppose A664: x in L~ (f | k) ; :: thesis: x in L~ g

L~ g1 c= L~ g by GOBOARD2:11;
hence x in L~ g by A44, A664; :: thesis: verum

end;

suppose x in LSeg (f,k) ; :: thesis: x in L~ g

then consider p1 being Point of (TOP-REAL 2) such that
A665: p1 = x and
A666: p1 `1 = (G * (i1,i2)) `1 and
A667: (G * (i1,i2)) `2 <= p1 `2 and
A668: p1 `2 <= (G * (i1,j2)) `2 by A478;
defpred S₂[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds
q `2 <= p1 `2 ) );

A669: now

reconsider n = len g1 as Nat ;
take n = n; :: thesis: S₂[n]
thus S₂[n] :: thesis: verum

proof

thus ( len g1 <= n & n <= len g ) by A628, XREAL_1:33; :: thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds
q `2 <= p1 `2
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A670: len g1 in dom g1 by FINSEQ_3:27;
let q be Point of (TOP-REAL 2); :: thesis: ( q = g /. n implies q `2 <= p1 `2 )
assume q = g /. n ; :: thesis: q `2 <= p1 `2
then q = (f | k) /. (len (f | k)) by A46, A670, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence q `2 <= p1 `2 by A667; :: thesis: verum

end;

end;

A671: for n being Nat st S₂[n] holds
n <= len g ;
consider ma being Nat such that
A672: ( S₂[ma] & ( for n being Nat st S₂[n] holds
n <= ma ) ) from NAT_1:sch 6(A671, A669);
reconsider ma = ma as Element of NAT by ORDINAL1:def 13;

now

per cases ( ma = len g or ma <> len g ) ;

suppose A673: ma = len g ; :: thesis: x in L~ g

i2 + 1 <= j2 by A390, NAT_1:13;
then A674: 1 <= l by XREAL_1:21;
then (len g1) + 1 <= ma by A396, A628, A673, XREAL_1:9;
then A675: len g1 <= ma - 1 by XREAL_1:21;
then 0 + 1 <= ma by XREAL_1:21;
then reconsider m1 = ma - 1 as Element of NAT by INT_1:18;
reconsider q = g /. m1 as Point of (TOP-REAL 2) ;
A676: ma - 1 <= len g by A673, XREAL_1:45;
then A677: q `1 = (G * (i1,i2)) `1 by A629, A675;
A678: q `2 <= (G * (i1,j2)) `2 by A629, A676, A675;
set lq = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & q `2 <= e `2 & e `2 <= (G * (i1,j2)) `2 ) } ;
A679: i2 + l = j2 ;
A680: l in dom g2 by A396, A674, FINSEQ_3:27;
then A681: g /. ma = g2 /. l by A396, A628, A673, FINSEQ_4:84
.= G * (i1,j2) by A396, A680, A679 ;
then (G * (i1,j2)) `2 <= p1 `2 by A672;
then A682: p1 `2 = (G * (i1,j2)) `2 by A668, XXREAL_0:1;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A683: 1 <= m1 by A675, XXREAL_0:2;
A684: m1 + 1 = ma ;
then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg (q,(G * (i1,j2))) ) by A673, A681, A683, EUCLID:57, TOPREAL1:def 5;
then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & q `2 <= e `2 & e `2 <= (G * (i1,j2)) `2 ) } by A462, A477, A677, A678, TOPREAL3:15;
then A685: p1 in LSeg (g,m1) by A666, A682, A678;
LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A673, A683, A684;
hence x in L~ g by A665, A685, TARSKI:def 4; :: thesis: verum

end;

suppose ma <> len g ; :: thesis: x in L~ g

then ma < len g by A672, XXREAL_0:1;
then A686: ma + 1 <= len g by NAT_1:13;
reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ;
set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa `2 <= p2 `2 & p2 `2 <= qa1 `2 ) } ;
A687: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:57;
A688: qa `2 <= p1 `2 by A672;
A689: len g1 <= ma + 1 by A672, NAT_1:13;
then A690: qa1 `1 = (G * (i1,i2)) `1 by A629, A686;

A691: now

assume qa1 `2 <= p1 `2 ; :: thesis: contradiction
then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds
q `2 <= p1 `2 ;
then ma + 1 <= ma by A672, A686, A689;
hence contradiction by XREAL_1:31; :: thesis: verum

end;

A692: ( qa `1 = (G * (i1,i2)) `1 & qa = |[(qa `1),(qa `2)]| ) by A629, A672, EUCLID:57;
A693: 1 <= ma by A24, A14, A47, A672, NAT_1:13;
then LSeg (g,ma) = LSeg (qa,qa1) by A686, TOPREAL1:def 5
.= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa `2 <= p2 `2 & p2 `2 <= qa1 `2 ) } by A688, A691, A690, A692, A687, TOPREAL3:15, XXREAL_0:2 ;
then A694: x in LSeg (g,ma) by A665, A666, A688, A691;
LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A686, A693;
hence x in L~ g by A694, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then 1 in dom g1 by FINSEQ_3:27;
hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:83
.= f /. 1 by A27, A25, FINSEQ_4:86 ;
:: thesis: ( g /. (len g) = f /. (len f) & len f <= len g )
A695: len g = (len g1) + l by A396, FINSEQ_1:35;
i2 + 1 <= j2 by A390, NAT_1:13;
then A696: 1 <= l by XREAL_1:21;
then A697: l in dom g2 by A396, FINSEQ_3:27;
hence g /. (len g) = g2 /. l by A695, FINSEQ_4:84
.= G * (i1,(i2 + l)) by A396, A697
.= f /. (len f) by A3, A21, A76 ;
:: thesis: len f <= len g
thus len f <= len g by A3, A14, A47, A696, A695, XREAL_1:9; :: thesis: verum

end;

end;

end;

hence ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; :: thesis: verum

end;

suppose A698: i2 = j2 ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )

set ppi = G * (i1,i2);
set pj = G * (j1,i2);

now

per cases ( i1 > j1 or i1 = j1 or i1 < j1 ) by XXREAL_0:1;

case A699: i1 > j1 ; :: thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

c1 /. i1 = c1 . i1 by A66, A60, PARTFUN1:def 8;
then A700: c1 /. i1 = G * (i1,i2) by A66, MATRIX_1:def 9;
then A701: (X_axis c1) . i1 = (G * (i1,i2)) `1 by A66, A18, A63, A64, A59, GOBOARD1:def 3;
c1 /. j1 = c1 . j1 by A23, A60, PARTFUN1:def 8;
then A702: c1 /. j1 = G * (j1,i2) by A23, MATRIX_1:def 9;
then A703: (X_axis c1) . j1 = (G * (j1,i2)) `1 by A23, A18, A63, A64, A59, GOBOARD1:def 3;
then A704: (G * (j1,i2)) `1 < (G * (i1,i2)) `1 by A66, A23, A18, A69, A63, A64, A59, A699, A701, SEQM_3:def 1;
reconsider l = i1 - j1 as Element of NAT by A699, INT_1:18;
defpred S₂[ Nat, set ] means for m being Element of NAT st m = i1 - $1 holds
$2 = G * (m,i2);
set lk = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } ;
A705: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:57;

A706: now

let n be Element of NAT ; :: thesis: ( n in Seg l implies ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G ) )
assume n in Seg l ; :: thesis: ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G )
then A707: n <= l by FINSEQ_1:3;
l <= i1 by XREAL_1:45;
then reconsider w = i1 - n as Element of NAT by A707, INT_1:18, XXREAL_0:2;
( i1 - n <= i1 & i1 <= len G ) by A66, FINSEQ_3:27, XREAL_1:45;
then A708: w <= len G by XXREAL_0:2;
A709: 1 <= j1 by A23, FINSEQ_3:27;
i1 - l <= i1 - n by A707, XREAL_1:15;
then 1 <= w by A709, XXREAL_0:2;
then w in dom G by A708, FINSEQ_3:27;
hence ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G ) by A22, A68, ZFMISC_1:106; :: thesis: verum

end;

A710: now

let n be Nat; :: thesis: ( n in Seg l implies ex p being Element of the U1 of (TOP-REAL 2) st S₂[n,p] )
assume n in Seg l ; :: thesis: ex p being Element of the U1 of (TOP-REAL 2) st S₂[n,p]
then reconsider m = i1 - n as Element of NAT by A706;
take p = G * (m,i2); :: thesis: S₂[n,p]
thus S₂[n,p] ; :: thesis: verum

end;

consider g2 being FinSequence of (TOP-REAL 2) such that
A711: ( len g2 = l & ( for n being Nat st n in Seg l holds
S₂[n,g2 /. n] ) ) from FINSEQ_4:sch 1(A710);
take g = g1 ^ g2; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A712: dom g2 = Seg l by A711, FINSEQ_1:def 3;

now

let n be Element of NAT ; :: thesis: ( n in dom g2 implies ex m, k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) ) )
assume A713: n in dom g2 ; :: thesis: ex m, k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
then reconsider m = i1 - n as Element of NAT by A706, A712;
take m = m; :: thesis: ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
take k = i2; :: thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) )
thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A706, A711, A712, A713; :: thesis: verum

end;

then A714: for n being Element of NAT st n in dom g holds
ex i, j being Element of NAT st
( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:39;
A715: Seg (len g2) = dom g2 by FINSEQ_1:def 3;
A716: (Y_axis c1) . i1 = (G * (i1,i2)) `2 by A66, A18, A61, A59, A700, GOBOARD1:def 4;

A717: now

let n be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds
( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 )
let p be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & g2 /. n = p implies ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 ) )
assume that
A718: n in dom g2 and
A719: g2 /. n = p ; :: thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 )
reconsider n1 = i1 - n as Element of NAT by A706, A712, A718;
n <= len g2 by A715, A718, FINSEQ_1:3;
then A720: i1 - (len g2) <= n1 by XREAL_1:15;
set pn = G * (n1,i2);
A721: g2 /. n = G * (n1,i2) by A711, A715, A718;
A722: i1 - n in dom G by A706, A711, A715, A718;
then A723: (Y_axis c1) . n1 = (Y_axis c1) . i1 by A66, A18, A70, A61, A59, SEQM_3:def 15;
c1 /. n1 = c1 . n1 by A60, A722, PARTFUN1:def 8;
then A724: c1 /. n1 = G * (n1,i2) by A722, MATRIX_1:def 9;
then A725: (X_axis c1) . n1 = (G * (n1,i2)) `1 by A18, A63, A64, A59, A722, GOBOARD1:def 3;
(Y_axis c1) . n1 = (G * (n1,i2)) `2 by A18, A61, A59, A722, A724, GOBOARD1:def 4;
hence ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 ) by A66, A23, A18, A69, A63, A64, A59, A711, A716, A701, A703, A719, A722, A721, A720, A723, A725, SEQ_4:154, XREAL_1:45; :: thesis: ( p in rng c1 & p `1 < (G * (i1,i2)) `1 )
thus p in rng c1 by A60, A719, A722, A721, A724, PARTFUN2:4; :: thesis: p `1 < (G * (i1,i2)) `1
1 <= n by A715, A718, FINSEQ_1:3;
then n1 < i1 by XREAL_1:46;
hence p `1 < (G * (i1,i2)) `1 by A66, A18, A69, A63, A64, A59, A701, A719, A722, A721, A725, SEQM_3:def 1; :: thesis: verum

end;

A726: g2 is special

proof

let n be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
set p = g2 /. n;
assume A727: ( 1 <= n & n + 1 <= len g2 ) ; :: thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
then n in dom g2 by SEQ_4:151;
then A728: (g2 /. n) `2 = (G * (i1,i2)) `2 by A717;
n + 1 in dom g2 by A727, SEQ_4:151;
hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A717, A728; :: thesis: verum

end;

A729: now

let n, m be Element of NAT ; :: thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds
q `1 < p `1
let p, q be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies q `1 < p `1 )
assume that
A730: n in dom g2 and
A731: m in dom g2 and
A732: n < m and
A733: ( g2 /. n = p & g2 /. m = q ) ; :: thesis: q `1 < p `1
A734: i1 - n in dom G by A706, A712, A730;
reconsider n1 = i1 - n, m1 = i1 - m as Element of NAT by A706, A712, A730, A731;
set pn = G * (n1,i2);
set pm = G * (m1,i2);
A735: m1 < n1 by A732, XREAL_1:17;
c1 /. n1 = c1 . n1 by A60, A706, A712, A730, PARTFUN1:def 8;
then c1 /. n1 = G * (n1,i2) by A734, MATRIX_1:def 9;
then A736: (X_axis c1) . n1 = (G * (n1,i2)) `1 by A65, A60, A734, GOBOARD1:def 3;
A737: i1 - m in dom G by A706, A712, A731;
c1 /. m1 = c1 . m1 by A60, A706, A712, A731, PARTFUN1:def 8;
then c1 /. m1 = G * (m1,i2) by A737, MATRIX_1:def 9;
then A738: (X_axis c1) . m1 = (G * (m1,i2)) `1 by A65, A60, A737, GOBOARD1:def 3;
( g2 /. n = G * (n1,i2) & g2 /. m = G * (m1,i2) ) by A711, A712, A730, A731;
hence q `1 < p `1 by A69, A65, A60, A733, A734, A737, A735, A736, A738, SEQM_3:def 1; :: thesis: verum

end;

for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds
LSeg (g2,n) misses LSeg (g2,m)

proof

let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) )
assume that
A739: m > n + 1 and
A740: n in dom g2 and
A741: n + 1 in dom g2 and
A742: m in dom g2 and
A743: m + 1 in dom g2 and
A744: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A745: ( p1 `2 = (G * (i1,i2)) `2 & p2 `2 = (G * (i1,i2)) `2 ) by A717, A740, A741;
n < n + 1 by NAT_1:13;
then A746: p2 `1 < p1 `1 by A729, A740, A741;
set lp = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p2 `1 <= w `1 & w `1 <= p1 `1 ) } ;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } ;
A747: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:57;
m < m + 1 by NAT_1:13;
then A748: q2 `1 < q1 `1 by A729, A742, A743;
A749: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
consider x being Element of (LSeg (g2,n)) /\ (LSeg (g2,m));
A750: x in LSeg (g2,n) by A744, XBOOLE_0:def 4;
A751: ( q1 `2 = (G * (i1,i2)) `2 & q2 `2 = (G * (i1,i2)) `2 ) by A717, A742, A743;
A752: x in LSeg (g2,m) by A744, XBOOLE_0:def 4;
( 1 <= m & m + 1 <= len g2 ) by A742, A743, FINSEQ_3:27;
then LSeg (g2,m) = LSeg (q2,q1) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } by A748, A751, A749, TOPREAL3:16 ;
then A753: ex tm being Point of (TOP-REAL 2) st
( tm = x & tm `2 = (G * (i1,i2)) `2 & q2 `1 <= tm `1 & tm `1 <= q1 `1 ) by A752;
( 1 <= n & n + 1 <= len g2 ) by A740, A741, FINSEQ_3:27;
then LSeg (g2,n) = LSeg (p2,p1) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p2 `1 <= w `1 & w `1 <= p1 `1 ) } by A746, A745, A747, TOPREAL3:16 ;
then A754: ex tn being Point of (TOP-REAL 2) st
( tn = x & tn `2 = (G * (i1,i2)) `2 & p2 `1 <= tn `1 & tn `1 <= p1 `1 ) by A750;
q1 `1 < p2 `1 by A729, A739, A741, A742;
hence contradiction by A754, A753, XXREAL_0:2; :: thesis: verum

end;

then A755: g2 is s.n.c. by GOBOARD2:6;
A756: not f /. k in L~ g2

proof

set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume f /. k in L~ g2 ; :: thesis: contradiction
then consider X being set such that
A757: f /. k in X and
A758: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A759: X = LSeg (g2,m) and
A760: ( 1 <= m & m + 1 <= len g2 ) by A758;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A761: m in dom g2 by A760, SEQ_4:151;
then A762: q1 `2 = (G * (i1,i2)) `2 by A717;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } ;
A763: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
A764: m + 1 in dom g2 by A760, SEQ_4:151;
then A765: q2 `2 = (G * (i1,i2)) `2 by A717;
m < m + 1 by NAT_1:13;
then A766: q2 `1 < q1 `1 by A729, A761, A764;
LSeg (g2,m) = LSeg (q2,q1) by A760, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } by A762, A765, A766, A763, TOPREAL3:16 ;
then ex p being Point of (TOP-REAL 2) st
( p = f /. k & p `2 = (G * (i1,i2)) `2 & q2 `1 <= p `1 & p `1 <= q1 `1 ) by A757, A759;
hence contradiction by A29, A717, A761; :: thesis: verum

end;

(Y_axis c1) . j1 = (G * (j1,i2)) `2 by A23, A18, A61, A59, A702, GOBOARD1:def 4;
then A767: (G * (i1,i2)) `2 = (G * (j1,i2)) `2 by A66, A23, A18, A70, A61, A59, A716, SEQM_3:def 15;

now

let n, m be Element of NAT ; :: thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m )
assume that
A768: ( n in dom g2 & m in dom g2 ) and
A769: n <> m ; :: thesis: not g2 /. n = g2 /. m
reconsider n1 = i1 - n, m1 = i1 - m as Element of NAT by A706, A712, A768;
A770: ( g2 /. n = G * (n1,i2) & g2 /. m = G * (m1,i2) ) by A711, A712, A768;
assume A771: g2 /. n = g2 /. m ; :: thesis: contradiction
( [(i1 - n),i2] in Indices G & [(i1 - m),i2] in Indices G ) by A706, A712, A768;
then n1 = m1 by A770, A771, GOBOARD1:21;
hence contradiction by A769; :: thesis: verum

end;

then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds
n = m ;
then A772: g2 is one-to-one by PARTFUN2:16;
reconsider m1 = i1 - l as Element of NAT ;
A773: G * (j1,i2) = |[((G * (j1,i2)) `1),((G * (j1,i2)) `2)]| by EUCLID:57;
A774: LSeg (f,k) = LSeg ((G * (j1,i2)),(G * (i1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A704, A767, A705, A773, TOPREAL3:16 ;
A775: rng g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g2 or x in LSeg (f,k) )
assume x in rng g2 ; :: thesis: x in LSeg (f,k)
then consider n being Element of NAT such that
A776: n in dom g2 and
A777: g2 /. n = x by PARTFUN2:4;
reconsider n1 = i1 - n as Element of NAT by A706, A711, A715, A776;
set pn = G * (n1,i2);
A778: g2 /. n = G * (n1,i2) by A711, A715, A776;
then A779: (G * (n1,i2)) `1 <= (G * (i1,i2)) `1 by A717, A776;
( (G * (n1,i2)) `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= (G * (n1,i2)) `1 ) by A717, A776, A778;
hence x in LSeg (f,k) by A774, A777, A778, A779; :: thesis: verum

end;

A780: now

let n be Element of NAT ; :: thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A781: n in dom g2 and
A782: n + 1 in dom g2 ; :: thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1
reconsider m1 = i1 - n, m2 = i1 - (n + 1) as Element of NAT by A706, A712, A781, A782;
let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A783: [l1,l2] in Indices G and
A784: [l3,l4] in Indices G and
A785: g2 /. n = G * (l1,l2) and
A786: g2 /. (n + 1) = G * (l3,l4) ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( [(i1 - (n + 1)),i2] in Indices G & g2 /. (n + 1) = G * (m2,i2) ) by A706, A711, A712, A782;
then A787: ( l3 = m2 & l4 = i2 ) by A784, A786, GOBOARD1:21;
( [(i1 - n),i2] in Indices G & g2 /. n = G * (m1,i2) ) by A706, A711, A712, A781;
then ( l1 = m1 & l2 = i2 ) by A783, A785, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs ((i1 - n) - (i1 - (n + 1)))) + 0 by A787, ABSVALUE:7
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

now

let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A788: [l1,l2] in Indices G and
A789: [l3,l4] in Indices G and
A790: g1 /. (len g1) = G * (l1,l2) and
A791: g2 /. 1 = G * (l3,l4) and
len g1 in dom g1 and
A792: 1 in dom g2 ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
reconsider m1 = i1 - 1 as Element of NAT by A706, A712, A792;
( [(i1 - 1),i2] in Indices G & g2 /. 1 = G * (m1,i2) ) by A706, A711, A712, A792;
then A793: ( l3 = m1 & l4 = i2 ) by A789, A791, GOBOARD1:21;
(f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:86;
then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A788, A790, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs (i1 - (i1 - 1))) + 0 by A793, ABSVALUE:7
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

then for n being Element of NAT st n in dom g & n + 1 in dom g holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A48, A780, GOBOARD1:40;
hence g is_sequence_on G by A714, GOBOARD1:def 11; :: thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A794: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def 5;
A795: not f /. k in rng g2

proof

assume f /. k in rng g2 ; :: thesis: contradiction
then consider n being Element of NAT such that
A796: n in dom g2 and
A797: g2 /. n = f /. k by PARTFUN2:4;
reconsider n1 = i1 - n as Element of NAT by A706, A711, A715, A796;
( [(i1 - n),i2] in Indices G & g2 /. n = G * (n1,i2) ) by A706, A711, A715, A796;
then A798: n1 = i1 by A28, A29, A797, GOBOARD1:21;
0 < n by A715, A796, FINSEQ_1:3;
hence contradiction by A798; :: thesis: verum

end;

(rng g1) /\ (rng g2) = {}

proof

consider x being Element of (rng g1) /\ (rng g2);
assume A799: not (rng g1) /\ (rng g2) = {} ; :: thesis: contradiction
then A800: x in rng g2 by XBOOLE_0:def 4;
A801: x in rng g1 by A799, XBOOLE_0:def 4;

now

per cases ( k = 1 or 1 < k ) by A24, XXREAL_0:1;

suppose k = 1 ; :: thesis: contradiction

hence contradiction by A52, A795, A801, A800, TARSKI:def 1; :: thesis: verum

end;

suppose 1 < k ; :: thesis: contradiction

then ( x in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A775, A801, A800, GOBOARD2:9, XBOOLE_0:def 4;
hence contradiction by A795, A800, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

then rng g1 misses rng g2 by XBOOLE_0:def 7;
hence g is one-to-one by A40, A772, FINSEQ_3:98; :: thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A802: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds
(LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}

proof

let n be Element of NAT ; :: thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} )
assume that
A803: 1 <= n and
A804: n + 2 <= len g2 ; :: thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}
A805: n + 1 in dom g2 by A803, A804, SEQ_4:152;
then g2 /. (n + 1) in rng g2 by PARTFUN2:4;
then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u1 being Point of (TOP-REAL 2) such that
A806: g2 /. (n + 1) = u1 and
A807: u1 `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u1 `1 and
u1 `1 <= (G * (i1,i2)) `1 ;
A808: n + 2 in dom g2 by A803, A804, SEQ_4:152;
then g2 /. (n + 2) in rng g2 by PARTFUN2:4;
then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u2 being Point of (TOP-REAL 2) such that
A809: g2 /. (n + 2) = u2 and
A810: u2 `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u2 `1 and
u2 `1 <= (G * (i1,i2)) `1 ;
( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11;
then A811: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A804, A806, A809, TOPREAL1:def 5;
n + 1 < (n + 1) + 1 by NAT_1:13;
then A812: u2 `1 < u1 `1 by A729, A805, A808, A806, A809;
A813: n in dom g2 by A803, A804, SEQ_4:152;
then g2 /. n in rng g2 by PARTFUN2:4;
then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u being Point of (TOP-REAL 2) such that
A814: g2 /. n = u and
A815: u `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u `1 and
u `1 <= (G * (i1,i2)) `1 ;
n + 1 <= n + 2 by XREAL_1:8;
then n + 1 <= len g2 by A804, XXREAL_0:2;
then A816: LSeg (g2,n) = LSeg (u,u1) by A803, A814, A806, TOPREAL1:def 5;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } ;
n < n + 1 by NAT_1:13;
then A817: u1 `1 < u `1 by A729, A813, A805, A814, A806;
then A818: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } by A807, A812;
( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:57;
then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } by A814, A815, A809, A810, A812, A817, TOPREAL3:16, XXREAL_0:2;
hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A814, A806, A809, A816, A811, A818, TOPREAL1:14; :: thesis: verum

end;

thus g is unfolded :: thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

let n be Nat; :: according to TOPREAL1:def 8 :: thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} )
assume that
A819: 1 <= n and
A820: n + 2 <= len g ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}
A821: (n + 1) + 1 <= len g by A820;
n + 1 <= (n + 1) + 1 by NAT_1:11;
then A822: n + 1 <= len g by A820, XXREAL_0:2;
A823: len g = (len g1) + (len g2) by FINSEQ_1:35;
(n + 2) - (len g1) = (n - (len g1)) + 2 ;
then A824: (n - (len g1)) + 2 <= len g2 by A820, A823, XREAL_1:22;
A825: 1 <= n + 1 by NAT_1:11;
A826: n <= n + 1 by NAT_1:11;
A827: n + (1 + 1) = (n + 1) + 1 ;

per cases ( n + 2 <= len g1 or len g1 < n + 2 ) ;

suppose A828: n + 2 <= len g1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

A829: n + (1 + 1) = (n + 1) + 1 ;
A830: n + 1 in dom g1 by A819, A828, SEQ_4:152;
then A831: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:83;
n in dom g1 by A819, A828, SEQ_4:152;
then A832: LSeg (g1,n) = LSeg (g,n) by A830, TOPREAL3:25;
n + 2 in dom g1 by A819, A828, SEQ_4:152;
then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A830, A829, TOPREAL3:25;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A819, A828, A832, A831, TOPREAL1:def 8; :: thesis: verum

end;

suppose len g1 < n + 2 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then (len g1) + 1 <= n + 2 by NAT_1:13;
then A833: len g1 <= (n + 2) - 1 by XREAL_1:21;
thus (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} :: thesis: verum

proof

per cases ( len g1 = n + 1 or len g1 <> n + 1 ) ;

suppose A834: len g1 = n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then 1 <= (len g) - (len g1) by A821, XREAL_1:21;
then 1 in dom g2 by A823, FINSEQ_3:27;
then A835: g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u1 being Point of (TOP-REAL 2) such that
A836: g2 /. 1 = u1 and
u1 `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u1 `1 and
u1 `1 <= (G * (i1,i2)) `1 ;
G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (j1,i2))) by RLTOPSP1:69;
then A837: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A794, A775, A835, A836, TOPREAL1:12;
1 <= n + 1 by NAT_1:11;
then A838: n + 1 in dom g1 by A834, FINSEQ_3:27;
then A839: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A834, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;

now

1 < len g1 by A819, A834, NAT_1:13;
then A840: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A840, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A841: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A842: LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:26;
n in dom g1 by A819, A826, A834, FINSEQ_3:27;
then A843: LSeg (g,n) = LSeg (g1,n) by A838, TOPREAL3:25;
( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A819, A820, A825, A822, A827, TOPREAL1:27;
then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def 4;
then A844: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:37;
1 <= len g2 by A820, A827, A823, A834, XREAL_1:8;
then g /. (n + 2) = g2 /. 1 by A827, A834, SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A820, A825, A827, A839, A836, TOPREAL1:def 5;
then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A842, A841, A843, A839, A837, XBOOLE_1:27;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A844, XBOOLE_0:def 10; :: thesis: verum

end;

suppose len g1 <> n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then len g1 < n + 1 by A833, XXREAL_0:1;
then A845: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;
thus (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} :: thesis: verum

proof

per cases ( len g1 = n or len g1 <> n ) ;

suppose A846: len g1 = n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A847: 2 <= len g2 by A820, A823, XREAL_1:8;
then 1 <= len g2 by XXREAL_0:2;
then A848: g /. (n + 1) = g2 /. 1 by A846, SEQ_4:153;
1 <= len g2 by A847, XXREAL_0:2;
then A849: 1 in dom g2 by FINSEQ_3:27;
then g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u1 being Point of (TOP-REAL 2) such that
A850: g2 /. 1 = u1 and
A851: u1 `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u1 `1 and
A852: u1 `1 <= (G * (i1,i2)) `1 ;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then g /. n = (f | k) /. (len (f | k)) by A46, A846, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
then A853: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A819, A822, A848, A850, TOPREAL1:def 5;
A854: 2 in dom g2 by A847, FINSEQ_3:27;
then g2 /. 2 in rng g2 by PARTFUN2:4;
then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775;
then consider u2 being Point of (TOP-REAL 2) such that
A855: g2 /. 2 = u2 and
A856: u2 `2 = (G * (i1,i2)) `2 and
(G * (j1,i2)) `1 <= u2 `1 and
A857: u2 `1 <= (G * (i1,i2)) `1 ;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } ;
u2 = |[(u2 `1),(u2 `2)]| by EUCLID:57;
then A858: LSeg ((G * (i1,i2)),(g2 /. 2)) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A705, A855, A856, A857, TOPREAL3:16;
u2 `1 < u1 `1 by A729, A849, A854, A850, A855;
then A859: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A851, A852;
g /. (n + 2) = g2 /. 2 by A846, A847, SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg (u1,u2) by A820, A825, A827, A848, A850, A855, TOPREAL1:def 5;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A848, A850, A855, A859, A853, A858, TOPREAL1:14; :: thesis: verum

end;

suppose len g1 <> n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A860: len g1 < n by A845, XXREAL_0:1;
then (len g1) + 1 <= n by NAT_1:13;
then A861: 1 <= n1 by XREAL_1:21;
n1 + (len g1) = n ;
then A862: LSeg (g,n) = LSeg (g2,n1) by A822, A860, GOBOARD2:10;
A863: n + 1 = (n1 + 1) + (len g1) ;
(n1 + 1) + (len g1) = n + 1 ;
then n1 + 1 <= len g2 by A822, A823, XREAL_1:8;
then A864: g /. (n + 1) = g2 /. (n1 + 1) by A863, NAT_1:11, SEQ_4:153;
len g1 < n + 1 by A826, A860, XXREAL_0:2;
then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A821, A863, GOBOARD2:10;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A802, A824, A862, A864, A861; :: thesis: verum

end;

end;

end;

end;

end;

end;

end;

end;

end;

A865: L~ g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g2 or x in LSeg (f,k) )
set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume x in L~ g2 ; :: thesis: x in LSeg (f,k)
then consider X being set such that
A866: x in X and
A867: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A868: X = LSeg (g2,m) and
A869: ( 1 <= m & m + 1 <= len g2 ) by A867;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A870: LSeg (g2,m) = LSeg (q1,q2) by A869, TOPREAL1:def 5;
m + 1 in dom g2 by A869, SEQ_4:151;
then A871: g2 /. (m + 1) in rng g2 by PARTFUN2:4;
m in dom g2 by A869, SEQ_4:151;
then g2 /. m in rng g2 by PARTFUN2:4;
then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (j1,i2))) by A794, A775, A871, TOPREAL1:12;
hence x in LSeg (f,k) by A794, A866, A868, A870; :: thesis: verum

end;

A872: (L~ g1) /\ (L~ g2) = {}

proof

per cases ( k = 1 or k <> 1 ) ;

suppose k = 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

hence (L~ g1) /\ (L~ g2) = {} by A52; :: thesis: verum

end;

suppose k <> 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

then 1 < k by A24, XXREAL_0:1;
then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:9;
then A873: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A865, XBOOLE_1:26;

now

consider x being Element of (L~ g1) /\ (L~ g2);
assume (L~ g1) /\ (L~ g2) <> {} ; :: thesis: contradiction
then ( x in {(f /. k)} & x in L~ g2 ) by A873, TARSKI:def 3, XBOOLE_0:def 4;
hence contradiction by A756, TARSKI:def 1; :: thesis: verum

end;

hence (L~ g1) /\ (L~ g2) = {} ; :: thesis: verum

end;

end;

end;

for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds
LSeg (g,n) misses LSeg (g,m)

proof

A874: 1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A875: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:86 ;
reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ;
set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ;
set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ;
let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) )
assume that
A876: m > n + 1 and
A877: n in dom g and
A878: n + 1 in dom g and
A879: m in dom g and
A880: m + 1 in dom g ; :: thesis: LSeg (g,n) misses LSeg (g,m)
A881: 1 <= n by A877, FINSEQ_3:27;
j1 + 1 <= i1 by A699, NAT_1:13;
then A882: 1 <= l by XREAL_1:21;
then A883: 1 in dom g2 by A711, FINSEQ_3:27;
then A884: ( qq `2 = (G * (i1,i2)) `2 & qq `1 < (G * (i1,i2)) `1 ) by A717;
A885: g /. ((len g1) + 1) = qq by A711, A882, SEQ_4:153;
A886: (G * (j1,i2)) `1 <= qq `1 by A717, A883;
A887: m + 1 <= len g by A880, FINSEQ_3:27;
A888: 1 <= m + 1 by A880, FINSEQ_3:27;
A889: 1 <= n + 1 by A878, FINSEQ_3:27;
A890: n + 1 <= len g by A878, FINSEQ_3:27;
A891: qq = |[(qq `1),(qq `2)]| by EUCLID:57;
A892: 1 <= m by A879, FINSEQ_3:27;
set ql = { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & qq `1 <= z `1 & z `1 <= (G * (i1,i2)) `1 ) } ;
A893: n <= n + 1 by NAT_1:11;
A894: len g = (len g1) + (len g2) by FINSEQ_1:35;
then (len g1) + 1 <= len g by A711, A882, XREAL_1:9;
then A895: LSeg (g,(len g1)) = LSeg (qq,(G * (i1,i2))) by A874, A875, A885, TOPREAL1:def 5
.= { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & qq `1 <= z `1 & z `1 <= (G * (i1,i2)) `1 ) } by A705, A884, A891, TOPREAL3:16 ;
A896: m <= m + 1 by NAT_1:11;
then A897: n + 1 <= m + 1 by A876, XXREAL_0:2;

now

per cases ( m + 1 <= len g1 or len g1 < m + 1 ) ;

suppose A898: m + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then m <= len g1 by A896, XXREAL_0:2;
then A899: m in dom g1 by A892, FINSEQ_3:27;
m + 1 in dom g1 by A888, A898, FINSEQ_3:27;
then A900: LSeg (g,m) = LSeg (g1,m) by A899, TOPREAL3:25;
A901: n + 1 <= len g1 by A897, A898, XXREAL_0:2;
then n <= len g1 by A893, XXREAL_0:2;
then A902: n in dom g1 by A881, FINSEQ_3:27;
n + 1 in dom g1 by A889, A901, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A902, TOPREAL3:25;
hence LSeg (g,n) misses LSeg (g,m) by A42, A876, A900, TOPREAL1:def 9; :: thesis: verum

end;

suppose len g1 < m + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A903: len g1 <= m by NAT_1:13;
then reconsider m1 = m - (len g1) as Element of NAT by INT_1:18;

now

per cases ( m = len g1 or m <> len g1 ) ;

suppose A904: m = len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

A905: LSeg (g,m) c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) )
assume x in LSeg (g,m) ; :: thesis: x in LSeg (f,k)
then consider px being Point of (TOP-REAL 2) such that
A906: ( px = x & px `2 = (G * (i1,i2)) `2 ) and
A907: qq `1 <= px `1 and
A908: px `1 <= (G * (i1,i2)) `1 by A895, A904;
(G * (j1,i2)) `1 <= px `1 by A886, A907, XXREAL_0:2;
hence x in LSeg (f,k) by A774, A906, A908; :: thesis: verum

end;

n <= len g1 by A876, A893, A904, XXREAL_0:2;
then A909: n in dom g1 by A881, FINSEQ_3:27;

now

1 < len g1 by A876, A889, A904, XXREAL_0:2;
then A910: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A910, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A911: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A912: n + 1 in dom g1 by A876, A889, A904, FINSEQ_3:27;
then A913: LSeg (g,n) = LSeg (g1,n) by A909, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A876, A881, A904;
then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:92;
then A914: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A911, A905, XBOOLE_1:27;

now

consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
assume A915: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A916: x in LSeg (g,n) by XBOOLE_0:def 4;
x in {(f /. k)} by A914, A915, TARSKI:def 3;
then A917: x = f /. k by TARSKI:def 1;
f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:86;
hence contradiction by A40, A41, A42, A876, A904, A909, A912, A913, A916, A917, GOBOARD2:7; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose m <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A918: len g1 < m by A903, XXREAL_0:1;
then (len g1) + 1 <= m by NAT_1:13;
then A919: 1 <= m1 by XREAL_1:21;
m + 1 = (m1 + 1) + (len g1) ;
then A920: m1 + 1 <= len g2 by A887, A894, XREAL_1:8;
m = m1 + (len g1) ;
then A921: LSeg (g,m) = LSeg (g2,m1) by A887, A918, GOBOARD2:10;
then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A919, A920;
then A922: LSeg (g,m) c= L~ g2 by ZFMISC_1:92;

now

per cases ( n + 1 <= len g1 or len g1 < n + 1 ) ;

suppose A923: n + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then n <= len g1 by A893, XXREAL_0:2;
then A924: n in dom g1 by A881, FINSEQ_3:27;
n + 1 in dom g1 by A889, A923, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A924, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A881, A923;
then LSeg (g,n) c= L~ g1 by ZFMISC_1:92;
then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A872, A922, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose len g1 < n + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A925: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;
A926: (n - (len g1)) + 1 = (n + 1) - (len g1) ;
A927: n = n1 + (len g1) ;

now

per cases ( len g1 = n or n <> len g1 ) ;

suppose A928: len g1 = n ; :: thesis: LSeg (g,n) misses LSeg (g,m)

now

reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ;
consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
set q1l = { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q2 `1 <= v `1 & v `1 <= q1 `1 ) } ;
A929: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
assume A930: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A931: x in LSeg (g,m) by XBOOLE_0:def 4;
x in LSeg (g,n) by A930, XBOOLE_0:def 4;
then A932: ex qx being Point of (TOP-REAL 2) st
( qx = x & qx `2 = (G * (i1,i2)) `2 & qq `1 <= qx `1 & qx `1 <= (G * (i1,i2)) `1 ) by A895, A928;
A933: m1 in dom g2 by A919, A920, SEQ_4:151;
then A934: q1 `2 = (G * (i1,i2)) `2 by A717;
A935: m1 + 1 in dom g2 by A919, A920, SEQ_4:151;
then A936: q2 `2 = (G * (i1,i2)) `2 by A717;
m1 < m1 + 1 by NAT_1:13;
then A937: q2 `1 < q1 `1 by A729, A933, A935;
LSeg (g2,m1) = LSeg (q2,q1) by A919, A920, TOPREAL1:def 5
.= { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q2 `1 <= v `1 & v `1 <= q1 `1 ) } by A934, A936, A937, A929, TOPREAL3:16 ;
then A938: ex qy being Point of (TOP-REAL 2) st
( qy = x & qy `2 = (G * (i1,i2)) `2 & q2 `1 <= qy `1 & qy `1 <= q1 `1 ) by A921, A931;
( m1 > n1 + 1 & n1 + 1 >= 1 ) by A876, A926, NAT_1:11, XREAL_1:11;
then m1 > 1 by XXREAL_0:2;
then q1 `1 < qq `1 by A729, A883, A933;
hence contradiction by A932, A938, XXREAL_0:2; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose n <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then len g1 < n by A925, XXREAL_0:1;
then A939: LSeg (g,n) = LSeg (g2,n1) by A890, A927, GOBOARD2:10;
m1 > n1 + 1 by A876, A926, XREAL_1:11;
hence LSeg (g,n) misses LSeg (g,m) by A755, A921, A939, TOPREAL1:def 9; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

hence g is s.n.c. by GOBOARD2:6; :: thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

now

set p = g1 /. (len g1);
set q = g2 /. 1;
j1 + 1 <= i1 by A699, NAT_1:13;
then 1 <= l by XREAL_1:21;
then 1 in dom g2 by A712, FINSEQ_1:3;
then (g2 /. 1) `2 = (G * (i1,i2)) `2 by A717;
hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:86; :: thesis: verum

end;

hence g is special by A43, A726, GOBOARD2:13; :: thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
thus L~ g = L~ f :: thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ;
set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ;
A940: len g = (len g1) + (len g2) by FINSEQ_1:35;

A941: now

let j be Element of NAT ; :: thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds
( b₃ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>3</sub> `1 & b₃ `1 <= (G * (i1,i2)) `1 & b₃ in rng c1 ) )
assume that
A942: len g1 <= j and
A943: j <= len g ; :: thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds
( b₃ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>3</sub> `1 & b₃ `1 <= (G * (i1,i2)) `1 & b₃ in rng c1 )
reconsider w = j - (len g1) as Element of NAT by A942, INT_1:18;
let p be Point of (TOP-REAL 2); :: thesis: ( p = g /. j implies ( b₂ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>2</sub> `1 & b₂ `1 <= (G * (i1,i2)) `1 & b₂ in rng c1 ) )
assume A944: p = g /. j ; :: thesis: ( b₂ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>2</sub> `1 & b₂ `1 <= (G * (i1,i2)) `1 & b₂ in rng c1 )

per cases ( j = len g1 or j <> len g1 ) ;

suppose A945: j = len g1 ; :: thesis: ( b₂ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>2</sub> `1 & b₂ `1 <= (G * (i1,i2)) `1 & b₂ in rng c1 )

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A946: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence p `2 = (G * (i1,i2)) `2 by A944, A945; :: thesis: ( (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 )
thus ( (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 ) by A66, A23, A18, A69, A63, A64, A59, A699, A701, A703, A944, A945, A946, SEQM_3:def 1; :: thesis: p in rng c1
Seg (len c1) = dom c1 by FINSEQ_1:def 3;
hence p in rng c1 by A66, A18, A59, A700, A944, A945, A946, PARTFUN2:4; :: thesis: verum

end;

suppose j <> len g1 ; :: thesis: ( b₂ `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b<sub>2</sub> `1 & b₂ `1 <= (G * (i1,i2)) `1 & b₂ in rng c1 )

then len g1 < j by A942, XXREAL_0:1;
then (len g1) + 1 <= j by NAT_1:13;
then A947: 1 <= w by XREAL_1:21;
A948: w <= len g2 by A940, A943, XREAL_1:22;
then A949: w in dom g2 by A947, FINSEQ_3:27;
j = w + (len g1) ;
then g /. j = g2 /. w by A947, A948, SEQ_4:153;
hence ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 ) by A717, A944, A949; :: thesis: verum

end;

end;

end;

thus L~ g c= L~ f :: according to XBOOLE_0:def 10 :: thesis: L~ f c= L~ g

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g or x in L~ f )
assume x in L~ g ; :: thesis: x in L~ f
then consider X being set such that
A950: x in X and
A951: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def 4;
consider i being Element of NAT such that
A952: X = LSeg (g,i) and
A953: 1 <= i and
A954: i + 1 <= len g by A951;

per cases ( i + 1 <= len g1 or i + 1 > len g1 ) ;

suppose A955: i + 1 <= len g1 ; :: thesis: x in L~ f

i <= i + 1 by NAT_1:11;
then i <= len g1 by A955, XXREAL_0:2;
then A956: i in dom g1 by A953, FINSEQ_3:27;
1 <= i + 1 by NAT_1:11;
then i + 1 in dom g1 by A955, FINSEQ_3:27;
then X = LSeg (g1,i) by A952, A956, TOPREAL3:25;
then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A953, A955;
then A957: x in L~ (f | k) by A44, A950, TARSKI:def 4;
L~ (f | k) c= L~ f by TOPREAL3:27;
hence x in L~ f by A957; :: thesis: verum

end;

suppose A958: i + 1 > len g1 ; :: thesis: x in L~ f

reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ;
A959: i <= len g by A954, NAT_1:13;
A960: len g1 <= i by A958, NAT_1:13;
then A961: q1 `2 = (G * (i1,i2)) `2 by A941, A959;
A962: q1 `1 <= (G * (i1,i2)) `1 by A941, A960, A959;
A963: (G * (j1,i2)) `1 <= q1 `1 by A941, A960, A959;
q2 `2 = (G * (i1,i2)) `2 by A941, A954, A958;
then A964: q2 = |[(q2 `1),(q1 `2)]| by A961, EUCLID:57;
A965: q2 `1 <= (G * (i1,i2)) `1 by A941, A954, A958;
A966: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A953, A954, EUCLID:57, TOPREAL1:def 5;
A967: (G * (j1,i2)) `1 <= q2 `1 by A941, A954, A958;

now

per cases ( q1 `1 > q2 `1 or q1 `1 = q2 `1 or q1 `1 < q2 `1 ) by XXREAL_0:1;

suppose q1 `1 > q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = q1 `2 & q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) } by A964, A966, TOPREAL3:16;
then consider p2 being Point of (TOP-REAL 2) such that
A968: ( p2 = x & p2 `2 = q1 `2 ) and
A969: ( q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) by A950, A952;
( (G * (j1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (i1,i2)) `1 ) by A962, A967, A969, XXREAL_0:2;
then A970: x in LSeg (f,k) by A774, A961, A968;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A970, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `1 = q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = {q1} by A964, A966, RLTOPSP1:71;
then x = q1 by A950, A952, TARSKI:def 1;
then A971: x in LSeg (f,k) by A774, A961, A963, A962;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A971, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `1 < q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `2 = q1 `2 & q1 `1 <= p1 `1 & p1 `1 <= q2 `1 ) } by A964, A966, TOPREAL3:16;
then consider p2 being Point of (TOP-REAL 2) such that
A972: ( p2 = x & p2 `2 = q1 `2 ) and
A973: ( q1 `1 <= p2 `1 & p2 `1 <= q2 `1 ) by A950, A952;
( (G * (j1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (i1,i2)) `1 ) by A963, A965, A973, XXREAL_0:2;
then A974: x in LSeg (f,k) by A774, A961, A972;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A974, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ f ; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ f or x in L~ g )
assume x in L~ f ; :: thesis: x in L~ g
then A975: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:8;

per cases ( x in L~ (f | k) or x in LSeg (f,k) ) by A975, XBOOLE_0:def 3;

suppose A976: x in L~ (f | k) ; :: thesis: x in L~ g

L~ g1 c= L~ g by GOBOARD2:11;
hence x in L~ g by A44, A976; :: thesis: verum

end;

suppose x in LSeg (f,k) ; :: thesis: x in L~ g

then consider p1 being Point of (TOP-REAL 2) such that
A977: p1 = x and
A978: p1 `2 = (G * (i1,i2)) `2 and
A979: (G * (j1,i2)) `1 <= p1 `1 and
A980: p1 `1 <= (G * (i1,i2)) `1 by A774;
defpred S₃[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds
q `1 >= p1 `1 ) );

A981: now

reconsider n = len g1 as Nat ;
take n = n; :: thesis: S₃[n]
thus S₃[n] :: thesis: verum

proof

thus ( len g1 <= n & n <= len g ) by A940, XREAL_1:33; :: thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds
q `1 >= p1 `1
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A982: len g1 in dom g1 by FINSEQ_3:27;
let q be Point of (TOP-REAL 2); :: thesis: ( q = g /. n implies q `1 >= p1 `1 )
assume q = g /. n ; :: thesis: q `1 >= p1 `1
then q = (f | k) /. (len (f | k)) by A46, A982, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence q `1 >= p1 `1 by A980; :: thesis: verum

end;

end;

A983: for n being Nat st S₃[n] holds
n <= len g ;
consider ma being Nat such that
A984: ( S₃[ma] & ( for n being Nat st S₃[n] holds
n <= ma ) ) from NAT_1:sch 6(A983, A981);
reconsider ma = ma as Element of NAT by ORDINAL1:def 13;

now

per cases ( ma = len g or ma <> len g ) ;

suppose A985: ma = len g ; :: thesis: x in L~ g

j1 + 1 <= i1 by A699, NAT_1:13;
then A986: 1 <= l by XREAL_1:21;
then (len g1) + 1 <= ma by A711, A940, A985, XREAL_1:9;
then A987: len g1 <= ma - 1 by XREAL_1:21;
then 0 + 1 <= ma by XREAL_1:21;
then reconsider m1 = ma - 1 as Element of NAT by INT_1:18;
reconsider q = g /. m1 as Point of (TOP-REAL 2) ;
A988: ma - 1 <= len g by A985, XREAL_1:45;
then A989: q `2 = (G * (i1,i2)) `2 by A941, A987;
A990: (G * (j1,i2)) `1 <= q `1 by A941, A988, A987;
set lq = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= e `1 & e `1 <= q `1 ) } ;
A991: i1 - l = j1 ;
A992: l in dom g2 by A711, A986, FINSEQ_3:27;
then A993: g /. ma = g2 /. l by A711, A940, A985, FINSEQ_4:84
.= G * (j1,i2) by A711, A712, A992, A991 ;
then p1 `1 <= (G * (j1,i2)) `1 by A984;
then A994: p1 `1 = (G * (j1,i2)) `1 by A979, XXREAL_0:1;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A995: 1 <= m1 by A987, XXREAL_0:2;
A996: m1 + 1 = ma ;
then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg ((G * (j1,i2)),q) ) by A985, A993, A995, EUCLID:57, TOPREAL1:def 5;
then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= e `1 & e `1 <= q `1 ) } by A767, A773, A989, A990, TOPREAL3:16;
then A997: p1 in LSeg (g,m1) by A978, A994, A990;
LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A985, A995, A996;
hence x in L~ g by A977, A997, TARSKI:def 4; :: thesis: verum

end;

suppose ma <> len g ; :: thesis: x in L~ g

then ma < len g by A984, XXREAL_0:1;
then A998: ma + 1 <= len g by NAT_1:13;
reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ;
set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa1 `1 <= p2 `1 & p2 `1 <= qa `1 ) } ;
A999: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:57;
A1000: p1 `1 <= qa `1 by A984;
A1001: len g1 <= ma + 1 by A984, NAT_1:13;
then A1002: qa1 `2 = (G * (i1,i2)) `2 by A941, A998;

A1003: now

assume p1 `1 <= qa1 `1 ; :: thesis: contradiction
then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds
p1 `1 <= q `1 ;
then ma + 1 <= ma by A984, A998, A1001;
hence contradiction by XREAL_1:31; :: thesis: verum

end;

A1004: ( qa `2 = (G * (i1,i2)) `2 & qa = |[(qa `1),(qa `2)]| ) by A941, A984, EUCLID:57;
A1005: 1 <= ma by A24, A14, A47, A984, NAT_1:13;
then LSeg (g,ma) = LSeg (qa1,qa) by A998, TOPREAL1:def 5
.= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa1 `1 <= p2 `1 & p2 `1 <= qa `1 ) } by A1000, A1003, A1002, A1004, A999, TOPREAL3:16, XXREAL_0:2 ;
then A1006: x in LSeg (g,ma) by A977, A978, A1000, A1003;
LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A998, A1005;
hence x in L~ g by A1006, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

end;

end;

A1007: len g = (len g1) + (len g2) by FINSEQ_1:35;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then 1 in dom g1 by FINSEQ_3:27;
hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:83
.= f /. 1 by A27, A25, FINSEQ_4:86 ;
:: thesis: ( g /. (len g) = f /. (len f) & len f <= len g )
j1 + 1 <= i1 by A699, NAT_1:13;
then A1008: 1 <= l by XREAL_1:21;
then A1009: l in dom g2 by A712, FINSEQ_1:3;
hence g /. (len g) = g2 /. l by A711, A1007, FINSEQ_4:84
.= G * (m1,i2) by A711, A712, A1009
.= f /. (len f) by A3, A21, A698 ;
:: thesis: len f <= len g
thus len f <= len g by A3, A14, A47, A711, A1008, A1007, XREAL_1:9; :: thesis: verum

end;

case A1010: i1 = j1 ; :: thesis: contradiction

k <> k + 1 ;
hence contradiction by A5, A27, A29, A19, A21, A698, A1010, PARTFUN2:17; :: thesis: verum

end;

case A1011: i1 < j1 ; :: thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

c1 /. i1 = c1 . i1 by A66, A60, PARTFUN1:def 8;
then A1012: c1 /. i1 = G * (i1,i2) by A66, MATRIX_1:def 9;
then A1013: (X_axis c1) . i1 = (G * (i1,i2)) `1 by A66, A65, A60, GOBOARD1:def 3;
c1 /. j1 = c1 . j1 by A23, A60, PARTFUN1:def 8;
then A1014: c1 /. j1 = G * (j1,i2) by A23, MATRIX_1:def 9;
then A1015: (X_axis c1) . j1 = (G * (j1,i2)) `1 by A23, A65, A60, GOBOARD1:def 3;
then A1016: (G * (i1,i2)) `1 < (G * (j1,i2)) `1 by A66, A23, A69, A65, A60, A1011, A1013, SEQM_3:def 1;
reconsider l = j1 - i1 as Element of NAT by A1011, INT_1:18;
deffunc H₁( Nat) -> Element of the U1 of (TOP-REAL 2) = G * ((i1 + $1),i2);
consider g2 being FinSequence of (TOP-REAL 2) such that
A1017: ( len g2 = l & ( for n being Nat st n in dom g2 holds
g2 /. n = H₁(n) ) ) from FINSEQ_4:sch 2();
take g = g1 ^ g2; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

A1018: now

let n be Element of NAT ; :: thesis: ( n in Seg l implies ( i1 + n in dom G & [(i1 + n),i2] in Indices G ) )
A1019: n <= i1 + n by NAT_1:11;
assume A1020: n in Seg l ; :: thesis: ( i1 + n in dom G & [(i1 + n),i2] in Indices G )
then n <= l by FINSEQ_1:3;
then A1021: i1 + n <= l + i1 by XREAL_1:9;
j1 <= len G by A23, FINSEQ_3:27;
then A1022: i1 + n <= len G by A1021, XXREAL_0:2;
1 <= n by A1020, FINSEQ_1:3;
then 1 <= i1 + n by A1019, XXREAL_0:2;
hence i1 + n in dom G by A1022, FINSEQ_3:27; :: thesis: [(i1 + n),i2] in Indices G
hence [(i1 + n),i2] in Indices G by A22, A68, ZFMISC_1:106; :: thesis: verum

end;

A1023: Seg (len g2) = dom g2 by FINSEQ_1:def 3;

now

let n be Element of NAT ; :: thesis: ( n in dom g2 implies ex m being Element of NAT ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) ) )
assume A1024: n in dom g2 ; :: thesis: ex m being Element of NAT ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
take m = i1 + n; :: thesis: ex k being Element of NAT st
( [m,k] in Indices G & g2 /. n = G * (m,k) )
take k = i2; :: thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) )
thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A1017, A1018, A1023, A1024; :: thesis: verum

end;

then A1025: for n being Element of NAT st n in dom g holds
ex i, j being Element of NAT st
( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:39;
A1026: (Y_axis c1) . i1 = (G * (i1,i2)) `2 by A66, A63, A64, A65, A61, A60, A1012, GOBOARD1:def 4;

A1027: now

let n be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds
( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 )
let p be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & g2 /. n = p implies ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 ) )
assume that
A1028: n in dom g2 and
A1029: g2 /. n = p ; :: thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 )
A1030: g2 /. n = G * ((i1 + n),i2) by A1017, A1028;
set n1 = i1 + n;
set pn = G * ((i1 + n),i2);
A1031: i1 + n in dom G by A1017, A1018, A1023, A1028;
then A1032: (Y_axis c1) . (i1 + n) = (Y_axis c1) . i1 by A66, A70, A63, A64, A65, A61, A60, SEQM_3:def 15;
c1 /. (i1 + n) = c1 . (i1 + n) by A60, A1017, A1018, A1023, A1028, PARTFUN1:def 8;
then A1033: c1 /. (i1 + n) = G * ((i1 + n),i2) by A1031, MATRIX_1:def 9;
then A1034: (X_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `1 by A65, A60, A1031, GOBOARD1:def 3;
n <= len g2 by A1028, FINSEQ_3:27;
then A1035: i1 + n <= i1 + (len g2) by XREAL_1:9;
(Y_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `2 by A63, A64, A65, A61, A60, A1031, A1033, GOBOARD1:def 4;
hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 ) by A66, A23, A69, A65, A60, A1017, A1026, A1013, A1015, A1029, A1030, A1031, A1035, A1032, A1034, SEQ_4:154, XREAL_1:33; :: thesis: ( p in rng c1 & p `1 > (G * (i1,i2)) `1 )
thus p in rng c1 by A60, A1029, A1030, A1031, A1033, PARTFUN2:4; :: thesis: p `1 > (G * (i1,i2)) `1
1 <= n by A1028, FINSEQ_3:27;
then i1 < i1 + n by XREAL_1:31;
hence p `1 > (G * (i1,i2)) `1 by A66, A69, A65, A60, A1013, A1029, A1030, A1031, A1034, SEQM_3:def 1; :: thesis: verum

end;

A1036: g2 is special

proof

let n be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
set p = g2 /. n;
assume A1037: ( 1 <= n & n + 1 <= len g2 ) ; :: thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 )
then n in dom g2 by SEQ_4:151;
then A1038: (g2 /. n) `2 = (G * (i1,i2)) `2 by A1027;
n + 1 in dom g2 by A1037, SEQ_4:151;
hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A1027, A1038; :: thesis: verum

end;

now

let n, m be Element of NAT ; :: thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m )
assume that
A1039: ( n in dom g2 & m in dom g2 ) and
A1040: n <> m ; :: thesis: not g2 /. n = g2 /. m
A1041: ( g2 /. n = G * ((i1 + n),i2) & g2 /. m = G * ((i1 + m),i2) ) by A1017, A1039;
assume A1042: g2 /. n = g2 /. m ; :: thesis: contradiction
( [(i1 + n),i2] in Indices G & [(i1 + m),i2] in Indices G ) by A1017, A1018, A1023, A1039;
then i1 + n = i1 + m by A1041, A1042, GOBOARD1:21;
hence contradiction by A1040; :: thesis: verum

end;

then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds
n = m ;
then A1043: g2 is one-to-one by PARTFUN2:16;
set lk = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } ;
A1044: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:57;

A1045: now

let n, m be Element of NAT ; :: thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds
p `1 < q `1
let p, q be Point of (TOP-REAL 2); :: thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies p `1 < q `1 )
assume that
A1046: n in dom g2 and
A1047: m in dom g2 and
A1048: n < m and
A1049: ( g2 /. n = p & g2 /. m = q ) ; :: thesis: p `1 < q `1
A1050: i1 + n in dom G by A1017, A1018, A1023, A1046;
set n1 = i1 + n;
set m1 = i1 + m;
set pn = G * ((i1 + n),i2);
set pm = G * ((i1 + m),i2);
A1051: i1 + n < i1 + m by A1048, XREAL_1:10;
c1 /. (i1 + n) = c1 . (i1 + n) by A60, A1017, A1018, A1023, A1046, PARTFUN1:def 8;
then c1 /. (i1 + n) = G * ((i1 + n),i2) by A1050, MATRIX_1:def 9;
then A1052: (X_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `1 by A65, A60, A1050, GOBOARD1:def 3;
A1053: i1 + m in dom G by A1017, A1018, A1023, A1047;
c1 /. (i1 + m) = c1 . (i1 + m) by A60, A1017, A1018, A1023, A1047, PARTFUN1:def 8;
then c1 /. (i1 + m) = G * ((i1 + m),i2) by A1053, MATRIX_1:def 9;
then A1054: (X_axis c1) . (i1 + m) = (G * ((i1 + m),i2)) `1 by A65, A60, A1053, GOBOARD1:def 3;
( g2 /. n = G * ((i1 + n),i2) & g2 /. m = G * ((i1 + m),i2) ) by A1017, A1046, A1047;
hence p `1 < q `1 by A69, A65, A60, A1049, A1050, A1053, A1051, A1052, A1054, SEQM_3:def 1; :: thesis: verum

end;

for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds
LSeg (g2,n) misses LSeg (g2,m)

proof

let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) )
assume that
A1055: m > n + 1 and
A1056: n in dom g2 and
A1057: n + 1 in dom g2 and
A1058: m in dom g2 and
A1059: m + 1 in dom g2 and
A1060: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A1061: ( p1 `2 = (G * (i1,i2)) `2 & p2 `2 = (G * (i1,i2)) `2 ) by A1027, A1056, A1057;
n < n + 1 by NAT_1:13;
then A1062: p1 `1 < p2 `1 by A1045, A1056, A1057;
set lp = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p1 `1 <= w `1 & w `1 <= p2 `1 ) } ;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } ;
A1063: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:57;
m < m + 1 by NAT_1:13;
then A1064: q1 `1 < q2 `1 by A1045, A1058, A1059;
A1065: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
consider x being Element of (LSeg (g2,n)) /\ (LSeg (g2,m));
A1066: x in LSeg (g2,n) by A1060, XBOOLE_0:def 4;
A1067: ( q1 `2 = (G * (i1,i2)) `2 & q2 `2 = (G * (i1,i2)) `2 ) by A1027, A1058, A1059;
A1068: x in LSeg (g2,m) by A1060, XBOOLE_0:def 4;
( 1 <= m & m + 1 <= len g2 ) by A1058, A1059, FINSEQ_3:27;
then LSeg (g2,m) = LSeg (q1,q2) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } by A1064, A1067, A1065, TOPREAL3:16 ;
then A1069: ex tm being Point of (TOP-REAL 2) st
( tm = x & tm `2 = (G * (i1,i2)) `2 & q1 `1 <= tm `1 & tm `1 <= q2 `1 ) by A1068;
( 1 <= n & n + 1 <= len g2 ) by A1056, A1057, FINSEQ_3:27;
then LSeg (g2,n) = LSeg (p1,p2) by TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p1 `1 <= w `1 & w `1 <= p2 `1 ) } by A1062, A1061, A1063, TOPREAL3:16 ;
then A1070: ex tn being Point of (TOP-REAL 2) st
( tn = x & tn `2 = (G * (i1,i2)) `2 & p1 `1 <= tn `1 & tn `1 <= p2 `1 ) by A1066;
p2 `1 < q1 `1 by A1045, A1055, A1057, A1058;
hence contradiction by A1070, A1069, XXREAL_0:2; :: thesis: verum

end;

then A1071: g2 is s.n.c. by GOBOARD2:6;
A1072: not f /. k in L~ g2

proof

set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume f /. k in L~ g2 ; :: thesis: contradiction
then consider X being set such that
A1073: f /. k in X and
A1074: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A1075: X = LSeg (g2,m) and
A1076: ( 1 <= m & m + 1 <= len g2 ) by A1074;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A1077: m in dom g2 by A1076, SEQ_4:151;
then A1078: q1 `2 = (G * (i1,i2)) `2 by A1027;
set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } ;
A1079: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
A1080: m + 1 in dom g2 by A1076, SEQ_4:151;
then A1081: q2 `2 = (G * (i1,i2)) `2 by A1027;
m < m + 1 by NAT_1:13;
then A1082: q1 `1 < q2 `1 by A1045, A1077, A1080;
LSeg (g2,m) = LSeg (q1,q2) by A1076, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } by A1078, A1081, A1082, A1079, TOPREAL3:16 ;
then ex p being Point of (TOP-REAL 2) st
( p = f /. k & p `2 = (G * (i1,i2)) `2 & q1 `1 <= p `1 & p `1 <= q2 `1 ) by A1073, A1075;
hence contradiction by A29, A1027, A1077; :: thesis: verum

end;

(Y_axis c1) . j1 = (G * (j1,i2)) `2 by A23, A63, A64, A65, A61, A60, A1014, GOBOARD1:def 4;
then A1083: (G * (i1,i2)) `2 = (G * (j1,i2)) `2 by A66, A23, A70, A63, A64, A65, A61, A60, A1026, SEQM_3:def 15;

A1084: now

let n be Element of NAT ; :: thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A1085: n in dom g2 and
A1086: n + 1 in dom g2 ; :: thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds
(abs (l1 - l3)) + (abs (l2 - l4)) = 1
let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A1087: [l1,l2] in Indices G and
A1088: [l3,l4] in Indices G and
A1089: g2 /. n = G * (l1,l2) and
A1090: g2 /. (n + 1) = G * (l3,l4) ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( g2 /. (n + 1) = G * ((i1 + (n + 1)),i2) & [(i1 + (n + 1)),i2] in Indices G ) by A1017, A1018, A1023, A1086;
then A1091: ( l3 = i1 + (n + 1) & l4 = i2 ) by A1088, A1090, GOBOARD1:21;
( g2 /. n = G * ((i1 + n),i2) & [(i1 + n),i2] in Indices G ) by A1017, A1018, A1023, A1085;
then ( l1 = i1 + n & l2 = i2 ) by A1087, A1089, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs ((i1 + n) - (i1 + (n + 1)))) + 0 by A1091, ABSVALUE:7
.= abs (- 1)
.= abs 1 by COMPLEX1:138
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

now

let l1, l2, l3, l4 be Element of NAT ; :: thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 )
assume that
A1092: [l1,l2] in Indices G and
A1093: [l3,l4] in Indices G and
A1094: g1 /. (len g1) = G * (l1,l2) and
A1095: g2 /. 1 = G * (l3,l4) and
len g1 in dom g1 and
A1096: 1 in dom g2 ; :: thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1
( g2 /. 1 = G * ((i1 + 1),i2) & [(i1 + 1),i2] in Indices G ) by A1017, A1018, A1023, A1096;
then A1097: ( l3 = i1 + 1 & l4 = i2 ) by A1093, A1095, GOBOARD1:21;
(f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:86;
then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A1092, A1094, GOBOARD1:21;
hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs (i1 - (i1 + 1))) + 0 by A1097, ABSVALUE:7
.= abs ((i1 - i1) + (- 1))
.= abs 1 by COMPLEX1:138
.= 1 by ABSVALUE:def 1 ;
:: thesis: verum

end;

then for n being Element of NAT st n in dom g & n + 1 in dom g holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A48, A1084, GOBOARD1:40;
hence g is_sequence_on G by A1025, GOBOARD1:def 11; :: thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A1098: G * (j1,i2) = |[((G * (j1,i2)) `1),((G * (j1,i2)) `2)]| by EUCLID:57;
A1099: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def 5
.= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1016, A1083, A1044, A1098, TOPREAL3:16 ;
A1100: rng g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng g2 or x in LSeg (f,k) )
assume x in rng g2 ; :: thesis: x in LSeg (f,k)
then consider n being Element of NAT such that
A1101: n in dom g2 and
A1102: g2 /. n = x by PARTFUN2:4;
set pn = G * ((i1 + n),i2);
A1103: g2 /. n = G * ((i1 + n),i2) by A1017, A1101;
then A1104: (G * ((i1 + n),i2)) `1 <= (G * (j1,i2)) `1 by A1027, A1101;
( (G * ((i1 + n),i2)) `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= (G * ((i1 + n),i2)) `1 ) by A1027, A1101, A1103;
hence x in LSeg (f,k) by A1099, A1102, A1103, A1104; :: thesis: verum

end;

A1105: Seg l = dom g2 by A1017, FINSEQ_1:def 3;
A1106: not f /. k in rng g2

proof

assume f /. k in rng g2 ; :: thesis: contradiction
then consider n being Element of NAT such that
A1107: n in dom g2 and
A1108: g2 /. n = f /. k by PARTFUN2:4;
( g2 /. n = G * ((i1 + n),i2) & [(i1 + n),i2] in Indices G ) by A1017, A1105, A1018, A1107;
then A1109: i1 + n = i1 by A28, A29, A1108, GOBOARD1:21;
0 < n by A1107, FINSEQ_3:27;
hence contradiction by A1109; :: thesis: verum

end;

(rng g1) /\ (rng g2) = {}

proof

consider x being Element of (rng g1) /\ (rng g2);
assume A1110: not (rng g1) /\ (rng g2) = {} ; :: thesis: contradiction
then A1111: x in rng g2 by XBOOLE_0:def 4;
A1112: x in rng g1 by A1110, XBOOLE_0:def 4;

now

per cases ( k = 1 or 1 < k ) by A24, XXREAL_0:1;

suppose k = 1 ; :: thesis: contradiction

hence contradiction by A52, A1106, A1112, A1111, TARSKI:def 1; :: thesis: verum

end;

suppose 1 < k ; :: thesis: contradiction

then ( x in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A1100, A1112, A1111, GOBOARD2:9, XBOOLE_0:def 4;
hence contradiction by A1106, A1111, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

then rng g1 misses rng g2 by XBOOLE_0:def 7;
hence g is one-to-one by A40, A1043, FINSEQ_3:98; :: thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
A1113: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def 5;
A1114: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds
(LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}

proof

let n be Element of NAT ; :: thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} )
assume that
A1115: 1 <= n and
A1116: n + 2 <= len g2 ; :: thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))}
A1117: n + 1 in dom g2 by A1115, A1116, SEQ_4:152;
then g2 /. (n + 1) in rng g2 by PARTFUN2:4;
then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u1 being Point of (TOP-REAL 2) such that
A1118: g2 /. (n + 1) = u1 and
A1119: u1 `2 = (G * (i1,i2)) `2 and
(G * (i1,i2)) `1 <= u1 `1 and
u1 `1 <= (G * (j1,i2)) `1 ;
A1120: n + 2 in dom g2 by A1115, A1116, SEQ_4:152;
then g2 /. (n + 2) in rng g2 by PARTFUN2:4;
then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u2 being Point of (TOP-REAL 2) such that
A1121: g2 /. (n + 2) = u2 and
A1122: u2 `2 = (G * (i1,i2)) `2 and
(G * (i1,i2)) `1 <= u2 `1 and
u2 `1 <= (G * (j1,i2)) `1 ;
( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11;
then A1123: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A1116, A1118, A1121, TOPREAL1:def 5;
n + 1 < (n + 1) + 1 by NAT_1:13;
then A1124: u1 `1 < u2 `1 by A1045, A1117, A1120, A1118, A1121;
A1125: n in dom g2 by A1115, A1116, SEQ_4:152;
then g2 /. n in rng g2 by PARTFUN2:4;
then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u being Point of (TOP-REAL 2) such that
A1126: g2 /. n = u and
A1127: u `2 = (G * (i1,i2)) `2 and
(G * (i1,i2)) `1 <= u `1 and
u `1 <= (G * (j1,i2)) `1 ;
n + 1 <= n + 2 by XREAL_1:8;
then n + 1 <= len g2 by A1116, XXREAL_0:2;
then A1128: LSeg (g2,n) = LSeg (u,u1) by A1115, A1126, A1118, TOPREAL1:def 5;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } ;
n < n + 1 by NAT_1:13;
then A1129: u `1 < u1 `1 by A1045, A1125, A1117, A1126, A1118;
then A1130: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } by A1119, A1124;
( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:57;
then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } by A1126, A1127, A1121, A1122, A1124, A1129, TOPREAL3:16, XXREAL_0:2;
hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A1126, A1118, A1121, A1128, A1123, A1130, TOPREAL1:14; :: thesis: verum

end;

thus g is unfolded :: thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

let n be Nat; :: according to TOPREAL1:def 8 :: thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} )
assume that
A1131: 1 <= n and
A1132: n + 2 <= len g ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}
A1133: (n + 1) + 1 <= len g by A1132;
n + 1 <= (n + 1) + 1 by NAT_1:11;
then A1134: n + 1 <= len g by A1132, XXREAL_0:2;
A1135: len g = (len g1) + (len g2) by FINSEQ_1:35;
(n + 2) - (len g1) = (n - (len g1)) + 2 ;
then A1136: (n - (len g1)) + 2 <= len g2 by A1132, A1135, XREAL_1:22;
A1137: 1 <= n + 1 by NAT_1:11;
A1138: n <= n + 1 by NAT_1:11;
A1139: n + (1 + 1) = (n + 1) + 1 ;

per cases ( n + 2 <= len g1 or len g1 < n + 2 ) ;

suppose A1140: n + 2 <= len g1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

A1141: n + (1 + 1) = (n + 1) + 1 ;
A1142: n + 1 in dom g1 by A1131, A1140, SEQ_4:152;
then A1143: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:83;
n in dom g1 by A1131, A1140, SEQ_4:152;
then A1144: LSeg (g1,n) = LSeg (g,n) by A1142, TOPREAL3:25;
n + 2 in dom g1 by A1131, A1140, SEQ_4:152;
then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A1142, A1141, TOPREAL3:25;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A1131, A1140, A1144, A1143, TOPREAL1:def 8; :: thesis: verum

end;

suppose len g1 < n + 2 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then (len g1) + 1 <= n + 2 by NAT_1:13;
then A1145: len g1 <= (n + 2) - 1 by XREAL_1:21;

now

per cases ( len g1 = n + 1 or len g1 <> n + 1 ) ;

suppose A1146: len g1 = n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then 1 <= (len g) - (len g1) by A1133, XREAL_1:21;
then 1 in dom g2 by A1135, FINSEQ_3:27;
then A1147: g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u1 being Point of (TOP-REAL 2) such that
A1148: g2 /. 1 = u1 and
u1 `2 = (G * (i1,i2)) `2 and
(G * (i1,i2)) `1 <= u1 `1 and
u1 `1 <= (G * (j1,i2)) `1 ;
G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (j1,i2))) by RLTOPSP1:69;
then A1149: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A1113, A1100, A1147, A1148, TOPREAL1:12;
1 <= n + 1 by NAT_1:11;
then A1150: n + 1 in dom g1 by A1146, FINSEQ_3:27;
then A1151: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A1146, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;

now

1 < len g1 by A1131, A1146, NAT_1:13;
then A1152: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A1152, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A1153: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A1154: LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:26;
n in dom g1 by A1131, A1138, A1146, FINSEQ_3:27;
then A1155: LSeg (g,n) = LSeg (g1,n) by A1150, TOPREAL3:25;
( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A1131, A1132, A1137, A1134, A1139, TOPREAL1:27;
then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def 4;
then A1156: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:37;
( n + 2 = 1 + (len g1) & 1 <= len g2 ) by A1132, A1139, A1135, A1146, XREAL_1:8;
then g /. (n + 2) = g2 /. 1 by SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A1132, A1137, A1139, A1151, A1148, TOPREAL1:def 5;
then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A1154, A1153, A1155, A1151, A1149, XBOOLE_1:27;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1156, XBOOLE_0:def 10; :: thesis: verum

end;

suppose len g1 <> n + 1 ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then len g1 < n + 1 by A1145, XXREAL_0:1;
then A1157: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;

now

per cases ( len g1 = n or len g1 <> n ) ;

suppose A1158: len g1 = n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A1159: 2 <= len g2 by A1132, A1135, XREAL_1:8;
then 1 <= len g2 by XXREAL_0:2;
then A1160: g /. (n + 1) = g2 /. 1 by A1158, SEQ_4:153;
1 <= len g2 by A1159, XXREAL_0:2;
then A1161: 1 in dom g2 by FINSEQ_3:27;
then g2 /. 1 in rng g2 by PARTFUN2:4;
then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u1 being Point of (TOP-REAL 2) such that
A1162: g2 /. 1 = u1 and
A1163: ( u1 `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= u1 `1 ) and
u1 `1 <= (G * (j1,i2)) `1 ;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then g /. n = (f | k) /. (len (f | k)) by A46, A1158, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
then A1164: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A1131, A1134, A1160, A1162, TOPREAL1:def 5;
A1165: 2 in dom g2 by A1159, FINSEQ_3:27;
then g2 /. 2 in rng g2 by PARTFUN2:4;
then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100;
then consider u2 being Point of (TOP-REAL 2) such that
A1166: g2 /. 2 = u2 and
A1167: ( u2 `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= u2 `1 ) and
u2 `1 <= (G * (j1,i2)) `1 ;
set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= u2 `1 ) } ;
u1 `1 < u2 `1 by A1045, A1161, A1165, A1162, A1166;
then ( u2 = |[(u2 `1),(u2 `2)]| & u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= u2 `1 ) } ) by A1163, EUCLID:57;
then A1168: u1 in LSeg ((G * (i1,i2)),u2) by A1044, A1167, TOPREAL3:16;
g /. (n + 2) = g2 /. 2 by A1158, A1159, SEQ_4:153;
then LSeg (g,(n + 1)) = LSeg (u1,u2) by A1132, A1137, A1139, A1160, A1162, A1166, TOPREAL1:def 5;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1160, A1162, A1164, A1168, TOPREAL1:14; :: thesis: verum

end;

suppose len g1 <> n ; :: thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))}

then A1169: len g1 < n by A1157, XXREAL_0:1;
then (len g1) + 1 <= n by NAT_1:13;
then A1170: 1 <= n1 by XREAL_1:21;
n1 + (len g1) = n ;
then A1171: LSeg (g,n) = LSeg (g2,n1) by A1134, A1169, GOBOARD2:10;
A1172: n + 1 = (n1 + 1) + (len g1) ;
then n1 + 1 <= len g2 by A1134, A1135, XREAL_1:8;
then A1173: g /. (n + 1) = g2 /. (n1 + 1) by A1172, NAT_1:11, SEQ_4:153;
len g1 < n + 1 by A1138, A1169, XXREAL_0:2;
then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A1133, A1172, GOBOARD2:10;
hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1114, A1136, A1171, A1173, A1170; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; :: thesis: verum

end;

end;

end;

A1174: L~ g2 c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g2 or x in LSeg (f,k) )
set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ;
assume x in L~ g2 ; :: thesis: x in LSeg (f,k)
then consider X being set such that
A1175: x in X and
A1176: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def 4;
consider m being Element of NAT such that
A1177: X = LSeg (g2,m) and
A1178: ( 1 <= m & m + 1 <= len g2 ) by A1176;
reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ;
A1179: LSeg (g2,m) = LSeg (q1,q2) by A1178, TOPREAL1:def 5;
m + 1 in dom g2 by A1178, SEQ_4:151;
then A1180: g2 /. (m + 1) in rng g2 by PARTFUN2:4;
m in dom g2 by A1178, SEQ_4:151;
then g2 /. m in rng g2 by PARTFUN2:4;
then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (j1,i2))) by A1113, A1100, A1180, TOPREAL1:12;
hence x in LSeg (f,k) by A1113, A1175, A1177, A1179; :: thesis: verum

end;

A1181: (L~ g1) /\ (L~ g2) = {}

proof

per cases ( k = 1 or k <> 1 ) ;

suppose k = 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

hence (L~ g1) /\ (L~ g2) = {} by A52; :: thesis: verum

end;

suppose k <> 1 ; :: thesis: (L~ g1) /\ (L~ g2) = {}

then 1 < k by A24, XXREAL_0:1;
then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:9;
then A1182: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A1174, XBOOLE_1:26;

now

consider x being Element of (L~ g1) /\ (L~ g2);
assume (L~ g1) /\ (L~ g2) <> {} ; :: thesis: contradiction
then ( x in {(f /. k)} & x in L~ g2 ) by A1182, TARSKI:def 3, XBOOLE_0:def 4;
hence contradiction by A1072, TARSKI:def 1; :: thesis: verum

end;

hence (L~ g1) /\ (L~ g2) = {} ; :: thesis: verum

end;

end;

end;

for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds
LSeg (g,n) misses LSeg (g,m)

proof

A1183: 1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A1184: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:86 ;
reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ;
set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ;
set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ;
let n, m be Element of NAT ; :: thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) )
assume that
A1185: m > n + 1 and
A1186: n in dom g and
A1187: n + 1 in dom g and
A1188: m in dom g and
A1189: m + 1 in dom g ; :: thesis: LSeg (g,n) misses LSeg (g,m)
A1190: 1 <= n by A1186, FINSEQ_3:27;
i1 + 1 <= j1 by A1011, NAT_1:13;
then A1191: 1 <= l by XREAL_1:21;
then A1192: 1 in dom g2 by A1017, FINSEQ_3:27;
then A1193: ( qq `2 = (G * (i1,i2)) `2 & qq `1 > (G * (i1,i2)) `1 ) by A1027;
A1194: g /. ((len g1) + 1) = qq by A1017, A1191, SEQ_4:153;
A1195: qq `1 <= (G * (j1,i2)) `1 by A1027, A1192;
A1196: m + 1 <= len g by A1189, FINSEQ_3:27;
A1197: 1 <= m + 1 by A1189, FINSEQ_3:27;
A1198: 1 <= n + 1 by A1187, FINSEQ_3:27;
A1199: n + 1 <= len g by A1187, FINSEQ_3:27;
A1200: qq = |[(qq `1),(qq `2)]| by EUCLID:57;
A1201: 1 <= m by A1188, FINSEQ_3:27;
set ql = { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= z `1 & z `1 <= qq `1 ) } ;
A1202: n <= n + 1 by NAT_1:11;
A1203: len g = (len g1) + (len g2) by FINSEQ_1:35;
then (len g1) + 1 <= len g by A1017, A1191, XREAL_1:9;
then A1204: LSeg (g,(len g1)) = LSeg ((G * (i1,i2)),qq) by A1183, A1184, A1194, TOPREAL1:def 5
.= { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= z `1 & z `1 <= qq `1 ) } by A1044, A1193, A1200, TOPREAL3:16 ;
A1205: m <= m + 1 by NAT_1:11;
then A1206: n + 1 <= m + 1 by A1185, XXREAL_0:2;

now

per cases ( m + 1 <= len g1 or len g1 < m + 1 ) ;

suppose A1207: m + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then m <= len g1 by A1205, XXREAL_0:2;
then A1208: m in dom g1 by A1201, FINSEQ_3:27;
m + 1 in dom g1 by A1197, A1207, FINSEQ_3:27;
then A1209: LSeg (g,m) = LSeg (g1,m) by A1208, TOPREAL3:25;
A1210: n + 1 <= len g1 by A1206, A1207, XXREAL_0:2;
then n <= len g1 by A1202, XXREAL_0:2;
then A1211: n in dom g1 by A1190, FINSEQ_3:27;
n + 1 in dom g1 by A1198, A1210, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A1211, TOPREAL3:25;
hence LSeg (g,n) misses LSeg (g,m) by A42, A1185, A1209, TOPREAL1:def 9; :: thesis: verum

end;

suppose len g1 < m + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A1212: len g1 <= m by NAT_1:13;
then reconsider m1 = m - (len g1) as Element of NAT by INT_1:18;

now

per cases ( m = len g1 or m <> len g1 ) ;

suppose A1213: m = len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

A1214: LSeg (g,m) c= LSeg (f,k)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) )
assume x in LSeg (g,m) ; :: thesis: x in LSeg (f,k)
then consider px being Point of (TOP-REAL 2) such that
A1215: ( px = x & px `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= px `1 ) and
A1216: px `1 <= qq `1 by A1204, A1213;
(G * (j1,i2)) `1 >= px `1 by A1195, A1216, XXREAL_0:2;
hence x in LSeg (f,k) by A1099, A1215; :: thesis: verum

end;

n <= len g1 by A1185, A1202, A1213, XXREAL_0:2;
then A1217: n in dom g1 by A1190, FINSEQ_3:27;

now

1 < len g1 by A1185, A1198, A1213, XXREAL_0:2;
then A1218: 1 + 1 <= len g1 by NAT_1:13;
assume k = 1 ; :: thesis: contradiction
hence contradiction by A52, A1218, TOPREAL1:29; :: thesis: verum

end;

then 1 < k by A24, XXREAL_0:1;
then A1219: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:9;
A1220: n + 1 in dom g1 by A1185, A1198, A1213, FINSEQ_3:27;
then A1221: LSeg (g,n) = LSeg (g1,n) by A1217, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A1185, A1190, A1213;
then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:92;
then A1222: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A1219, A1214, XBOOLE_1:27;

now

consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
assume A1223: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A1224: x in LSeg (g,n) by XBOOLE_0:def 4;
x in {(f /. k)} by A1222, A1223, TARSKI:def 3;
then A1225: x = f /. k by TARSKI:def 1;
f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:86;
hence contradiction by A40, A41, A42, A1185, A1213, A1217, A1220, A1221, A1224, A1225, GOBOARD2:7; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose m <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A1226: len g1 < m by A1212, XXREAL_0:1;
then (len g1) + 1 <= m by NAT_1:13;
then A1227: 1 <= m1 by XREAL_1:21;
m + 1 = (m1 + 1) + (len g1) ;
then A1228: m1 + 1 <= len g2 by A1196, A1203, XREAL_1:8;
m = m1 + (len g1) ;
then A1229: LSeg (g,m) = LSeg (g2,m1) by A1196, A1226, GOBOARD2:10;
then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A1227, A1228;
then A1230: LSeg (g,m) c= L~ g2 by ZFMISC_1:92;

now

per cases ( n + 1 <= len g1 or len g1 < n + 1 ) ;

suppose A1231: n + 1 <= len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then n <= len g1 by A1202, XXREAL_0:2;
then A1232: n in dom g1 by A1190, FINSEQ_3:27;
n + 1 in dom g1 by A1198, A1231, FINSEQ_3:27;
then LSeg (g,n) = LSeg (g1,n) by A1232, TOPREAL3:25;
then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A1190, A1231;
then LSeg (g,n) c= L~ g1 by ZFMISC_1:92;
then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A1181, A1230, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose len g1 < n + 1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then A1233: len g1 <= n by NAT_1:13;
then reconsider n1 = n - (len g1) as Element of NAT by INT_1:18;
A1234: (n - (len g1)) + 1 = (n + 1) - (len g1) ;
A1235: n = n1 + (len g1) ;

now

per cases ( len g1 = n or n <> len g1 ) ;

suppose A1236: len g1 = n ; :: thesis: LSeg (g,n) misses LSeg (g,m)

now

reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ;
consider x being Element of (LSeg (g,n)) /\ (LSeg (g,m));
set q1l = { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q1 `1 <= v `1 & v `1 <= q2 `1 ) } ;
A1237: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:57;
assume A1238: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; :: thesis: contradiction
then A1239: x in LSeg (g,m) by XBOOLE_0:def 4;
x in LSeg (g,n) by A1238, XBOOLE_0:def 4;
then A1240: ex qx being Point of (TOP-REAL 2) st
( qx = x & qx `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= qx `1 & qx `1 <= qq `1 ) by A1204, A1236;
A1241: m1 in dom g2 by A1227, A1228, SEQ_4:151;
then A1242: q1 `2 = (G * (i1,i2)) `2 by A1027;
A1243: m1 + 1 in dom g2 by A1227, A1228, SEQ_4:151;
then A1244: q2 `2 = (G * (i1,i2)) `2 by A1027;
m1 < m1 + 1 by NAT_1:13;
then A1245: q1 `1 < q2 `1 by A1045, A1241, A1243;
LSeg (g2,m1) = LSeg (q1,q2) by A1227, A1228, TOPREAL1:def 5
.= { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q1 `1 <= v `1 & v `1 <= q2 `1 ) } by A1242, A1244, A1245, A1237, TOPREAL3:16 ;
then A1246: ex qy being Point of (TOP-REAL 2) st
( qy = x & qy `2 = (G * (i1,i2)) `2 & q1 `1 <= qy `1 & qy `1 <= q2 `1 ) by A1229, A1239;
( m1 > n1 + 1 & n1 + 1 >= 1 ) by A1185, A1234, NAT_1:11, XREAL_1:11;
then m1 > 1 by XXREAL_0:2;
then qq `1 < q1 `1 by A1045, A1192, A1241;
hence contradiction by A1240, A1246, XXREAL_0:2; :: thesis: verum

end;

hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def 7; :: thesis: verum

end;

suppose n <> len g1 ; :: thesis: LSeg (g,n) misses LSeg (g,m)

then len g1 < n by A1233, XXREAL_0:1;
then A1247: LSeg (g,n) = LSeg (g2,n1) by A1199, A1235, GOBOARD2:10;
m1 > n1 + 1 by A1185, A1234, XREAL_1:11;
hence LSeg (g,n) misses LSeg (g,m) by A1071, A1229, A1247, TOPREAL1:def 9; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

end;

end;

hence LSeg (g,n) misses LSeg (g,m) ; :: thesis: verum

end;

hence g is s.n.c. by GOBOARD2:6; :: thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

now

set p = g1 /. (len g1);
set q = g2 /. 1;
i1 + 1 <= j1 by A1011, NAT_1:13;
then 1 <= l by XREAL_1:21;
then 1 in dom g2 by A1017, FINSEQ_3:27;
then (g2 /. 1) `2 = (G * (i1,i2)) `2 by A1027;
hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:86; :: thesis: verum

end;

hence g is special by A43, A1036, GOBOARD2:13; :: thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )
thus L~ g = L~ f :: thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof

set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ;
set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ;
A1248: len g = (len g1) + (len g2) by FINSEQ_1:35;

A1249: now

let j be Element of NAT ; :: thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds
( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) )
assume that
A1250: len g1 <= j and
A1251: j <= len g ; :: thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds
( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 )
reconsider w = j - (len g1) as Element of NAT by A1250, INT_1:18;
let p be Point of (TOP-REAL 2); :: thesis: ( p = g /. j implies ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) )
assume A1252: p = g /. j ; :: thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 )

now

per cases ( j = len g1 or j <> len g1 ) ;

suppose A1253: j = len g1 ; :: thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 )

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then len g1 in dom g1 by FINSEQ_3:27;
then A1254: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence p `2 = (G * (i1,i2)) `2 by A1252, A1253; :: thesis: ( (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 )
thus ( (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 ) by A66, A23, A69, A65, A60, A1011, A1013, A1015, A1252, A1253, A1254, SEQM_3:def 1; :: thesis: p in rng c1
thus p in rng c1 by A66, A60, A1012, A1252, A1253, A1254, PARTFUN2:4; :: thesis: verum

end;

suppose j <> len g1 ; :: thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 )

then len g1 < j by A1250, XXREAL_0:1;
then (len g1) + 1 <= j by NAT_1:13;
then A1255: 1 <= w by XREAL_1:21;
A1256: w <= len g2 by A1248, A1251, XREAL_1:22;
then A1257: w in dom g2 by A1255, FINSEQ_3:27;
j = w + (len g1) ;
then g /. j = g2 /. w by A1255, A1256, SEQ_4:153;
hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) by A1027, A1252, A1257; :: thesis: verum

end;

end;

end;

hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) ; :: thesis: verum

end;

thus L~ g c= L~ f :: according to XBOOLE_0:def 10 :: thesis: L~ f c= L~ g

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ g or x in L~ f )
assume x in L~ g ; :: thesis: x in L~ f
then consider X being set such that
A1258: x in X and
A1259: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def 4;
consider i being Element of NAT such that
A1260: X = LSeg (g,i) and
A1261: 1 <= i and
A1262: i + 1 <= len g by A1259;

now

per cases ( i + 1 <= len g1 or i + 1 > len g1 ) ;

suppose A1263: i + 1 <= len g1 ; :: thesis: x in L~ f

i <= i + 1 by NAT_1:11;
then i <= len g1 by A1263, XXREAL_0:2;
then A1264: i in dom g1 by A1261, FINSEQ_3:27;
1 <= i + 1 by NAT_1:11;
then i + 1 in dom g1 by A1263, FINSEQ_3:27;
then X = LSeg (g1,i) by A1260, A1264, TOPREAL3:25;
then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A1261, A1263;
then A1265: x in L~ (f | k) by A44, A1258, TARSKI:def 4;
L~ (f | k) c= L~ f by TOPREAL3:27;
hence x in L~ f by A1265; :: thesis: verum

end;

suppose A1266: i + 1 > len g1 ; :: thesis: x in L~ f

reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ;
A1267: i <= len g by A1262, NAT_1:13;
A1268: len g1 <= i by A1266, NAT_1:13;
then A1269: q1 `2 = (G * (i1,i2)) `2 by A1249, A1267;
A1270: q1 `1 <= (G * (j1,i2)) `1 by A1249, A1268, A1267;
A1271: (G * (i1,i2)) `1 <= q1 `1 by A1249, A1268, A1267;
q2 `2 = (G * (i1,i2)) `2 by A1249, A1262, A1266;
then A1272: q2 = |[(q2 `1),(q1 `2)]| by A1269, EUCLID:57;
A1273: q2 `1 <= (G * (j1,i2)) `1 by A1249, A1262, A1266;
A1274: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A1261, A1262, EUCLID:57, TOPREAL1:def 5;
A1275: (G * (i1,i2)) `1 <= q2 `1 by A1249, A1262, A1266;

now

per cases ( q1 `1 > q2 `1 or q1 `1 = q2 `1 or q1 `1 < q2 `1 ) by XXREAL_0:1;

suppose q1 `1 > q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = q1 `2 & q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) } by A1272, A1274, TOPREAL3:16;
then consider p2 being Point of (TOP-REAL 2) such that
A1276: ( p2 = x & p2 `2 = q1 `2 ) and
A1277: ( q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) by A1258, A1260;
( (G * (i1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (j1,i2)) `1 ) by A1270, A1275, A1277, XXREAL_0:2;
then A1278: x in LSeg (f,k) by A1099, A1269, A1276;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A1278, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `1 = q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = {q1} by A1272, A1274, RLTOPSP1:71;
then x = q1 by A1258, A1260, TARSKI:def 1;
then A1279: x in LSeg (f,k) by A1099, A1269, A1271, A1270;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A1279, TARSKI:def 4; :: thesis: verum

end;

suppose q1 `1 < q2 `1 ; :: thesis: x in L~ f

then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `2 = q1 `2 & q1 `1 <= p1 `1 & p1 `1 <= q2 `1 ) } by A1272, A1274, TOPREAL3:16;
then consider p2 being Point of (TOP-REAL 2) such that
A1280: ( p2 = x & p2 `2 = q1 `2 ) and
A1281: ( q1 `1 <= p2 `1 & p2 `1 <= q2 `1 ) by A1258, A1260;
( (G * (i1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (j1,i2)) `1 ) by A1271, A1273, A1281, XXREAL_0:2;
then A1282: x in LSeg (f,k) by A1099, A1269, A1280;
LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24;
hence x in L~ f by A1282, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ f ; :: thesis: verum

end;

end;

end;

hence x in L~ f ; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in L~ f or x in L~ g )
assume x in L~ f ; :: thesis: x in L~ g
then A1283: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:8;

now

per cases ( x in L~ (f | k) or x in LSeg (f,k) ) by A1283, XBOOLE_0:def 3;

suppose A1284: x in L~ (f | k) ; :: thesis: x in L~ g

L~ g1 c= L~ g by GOBOARD2:11;
hence x in L~ g by A44, A1284; :: thesis: verum

end;

suppose x in LSeg (f,k) ; :: thesis: x in L~ g

then consider p1 being Point of (TOP-REAL 2) such that
A1285: p1 = x and
A1286: p1 `2 = (G * (i1,i2)) `2 and
A1287: (G * (i1,i2)) `1 <= p1 `1 and
A1288: p1 `1 <= (G * (j1,i2)) `1 by A1099;
defpred S₂[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds
q `1 <= p1 `1 ) );

A1289: now

reconsider n = len g1 as Nat ;
take n = n; :: thesis: S₂[n]
thus S₂[n] :: thesis: verum

proof

thus ( len g1 <= n & n <= len g ) by A1248, XREAL_1:33; :: thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds
q `1 <= p1 `1
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A1290: len g1 in dom g1 by FINSEQ_3:27;
let q be Point of (TOP-REAL 2); :: thesis: ( q = g /. n implies q `1 <= p1 `1 )
assume q = g /. n ; :: thesis: q `1 <= p1 `1
then q = (f | k) /. (len (f | k)) by A46, A1290, FINSEQ_4:83
.= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:86 ;
hence q `1 <= p1 `1 by A1287; :: thesis: verum

end;

end;

A1291: for n being Nat st S₂[n] holds
n <= len g ;
consider ma being Nat such that
A1292: ( S₂[ma] & ( for n being Nat st S₂[n] holds
n <= ma ) ) from NAT_1:sch 6(A1291, A1289);
reconsider ma = ma as Element of NAT by ORDINAL1:def 13;

now

per cases ( ma = len g or ma <> len g ) ;

suppose A1293: ma = len g ; :: thesis: x in L~ g

i1 + 1 <= j1 by A1011, NAT_1:13;
then A1294: 1 <= l by XREAL_1:21;
then (len g1) + 1 <= ma by A1017, A1248, A1293, XREAL_1:9;
then A1295: len g1 <= ma - 1 by XREAL_1:21;
then 0 + 1 <= ma by XREAL_1:21;
then reconsider m1 = ma - 1 as Element of NAT by INT_1:18;
reconsider q = g /. m1 as Point of (TOP-REAL 2) ;
A1296: ma - 1 <= len g by A1293, XREAL_1:45;
then A1297: q `2 = (G * (i1,i2)) `2 by A1249, A1295;
A1298: q `1 <= (G * (j1,i2)) `1 by A1249, A1296, A1295;
set lq = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & q `1 <= e `1 & e `1 <= (G * (j1,i2)) `1 ) } ;
A1299: i1 + l = j1 ;
A1300: l in dom g2 by A1017, A1294, FINSEQ_3:27;
then A1301: g /. ma = g2 /. l by A1017, A1248, A1293, FINSEQ_4:84
.= G * (j1,i2) by A1017, A1300, A1299 ;
then (G * (j1,i2)) `1 <= p1 `1 by A1292;
then A1302: p1 `1 = (G * (j1,i2)) `1 by A1288, XXREAL_0:1;
1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then A1303: 1 <= m1 by A1295, XXREAL_0:2;
A1304: m1 + 1 = ma ;
then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg (q,(G * (j1,i2))) ) by A1293, A1301, A1303, EUCLID:57, TOPREAL1:def 5;
then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & q `1 <= e `1 & e `1 <= (G * (j1,i2)) `1 ) } by A1083, A1098, A1297, A1298, TOPREAL3:16;
then A1305: p1 in LSeg (g,m1) by A1286, A1302, A1298;
LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A1293, A1303, A1304;
hence x in L~ g by A1285, A1305, TARSKI:def 4; :: thesis: verum

end;

suppose ma <> len g ; :: thesis: x in L~ g

then ma < len g by A1292, XXREAL_0:1;
then A1306: ma + 1 <= len g by NAT_1:13;
reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ;
set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa `1 <= p2 `1 & p2 `1 <= qa1 `1 ) } ;
A1307: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:57;
A1308: qa `1 <= p1 `1 by A1292;
A1309: len g1 <= ma + 1 by A1292, NAT_1:13;
then A1310: qa1 `2 = (G * (i1,i2)) `2 by A1249, A1306;

A1311: now

assume qa1 `1 <= p1 `1 ; :: thesis: contradiction
then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds
q `1 <= p1 `1 ;
then ma + 1 <= ma by A1292, A1306, A1309;
hence contradiction by XREAL_1:31; :: thesis: verum

end;

A1312: ( qa `2 = (G * (i1,i2)) `2 & qa = |[(qa `1),(qa `2)]| ) by A1249, A1292, EUCLID:57;
A1313: 1 <= ma by A24, A14, A47, A1292, NAT_1:13;
then LSeg (g,ma) = LSeg (qa,qa1) by A1306, TOPREAL1:def 5
.= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa `1 <= p2 `1 & p2 `1 <= qa1 `1 ) } by A1308, A1311, A1310, A1312, A1307, TOPREAL3:16, XXREAL_0:2 ;
then A1314: x in LSeg (g,ma) by A1285, A1286, A1308, A1311;
LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A1306, A1313;
hence x in L~ g by A1314, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

end;

end;

hence x in L~ g ; :: thesis: verum

end;

1 <= len g1 by A24, A14, A47, XXREAL_0:2;
then 1 in dom g1 by FINSEQ_3:27;
hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:83
.= f /. 1 by A27, A25, FINSEQ_4:86 ;
:: thesis: ( g /. (len g) = f /. (len f) & len f <= len g )
A1315: len g = (len g1) + l by A1017, FINSEQ_1:35;
i1 + 1 <= j1 by A1011, NAT_1:13;
then A1316: 1 <= l by XREAL_1:21;
then A1317: l in dom g2 by A1017, FINSEQ_3:27;
hence g /. (len g) = g2 /. l by A1315, FINSEQ_4:84
.= G * ((i1 + l),i2) by A1017, A1317
.= f /. (len f) by A3, A21, A698 ;
:: thesis: len f <= len g
thus len f <= len g by A3, A14, A47, A1316, A1315, XREAL_1:9; :: thesis: verum

end;

end;

end;

hence ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; :: thesis: verum

end;

end;

end;

hence ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; :: thesis: verum

end;

end;

end;

A1318: S₁[ 0 ]

proof

let f be FinSequence of (TOP-REAL 2); :: thesis: ( len f = 0 & ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) )
assume that
A1319: len f = 0 and
A1320: ( ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special ) ; :: thesis: ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
take g = f; :: thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
f = {} by A1319;
then for n being Element of NAT st n in dom g & n + 1 in dom g holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by RELAT_1:60;
hence ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) by A1320, GOBOARD1:def 11; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A1318, A1);
hence ( ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) ; :: thesis: verum