let i, j, k be Element of NAT ; :: thesis:
let f be FinSequence of the carrier of (TOP-REAL 2); :: thesis:
let G be Go-board; :: thesis:
set X = (LSeg (G * i,j),(G * i,k)) /\ (L~ f);
assume that
A1: f is_sequence_on G and
A2: LSeg (G * i,j),(G * i,k) meets L~ f and
A3: [i,j] in Indices G and
A4: [i,k] in Indices G and
A5: j <= k ; :: thesis:
A6: ( 1 <= i & i <= len G ) by A3, MATRIX_1:39;
ex x being set st
( x in LSeg (G * i,j),(G * i,k) & x in L~ f ) by A2, XBOOLE_0:3;
then reconsider X1 = (LSeg (G * i,j),(G * i,k)) /\ (L~ f) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def 4;
consider p being set such that
A7: p in N-most X1 by XBOOLE_0:def 1;
reconsider p = p as Point of (TOP-REAL 2) by A7;
A8: p in (LSeg (G * i,j),(G * i,k)) /\ (L~ f) by A7, XBOOLE_0:def 4;
then A9: p in LSeg (G * i,j),(G * i,k) by XBOOLE_0:def 4;
p in L~ f by A8, XBOOLE_0:def 4;
then p in union { (LSeg f,k1) where k1 is Element of NAT : ( 1 <= k1 & k1 + 1 <= len f ) } by TOPREAL1:def 6;
then consider Y being set such that
A10: p in Y and
A11: Y in { (LSeg f,k1) where k1 is Element of NAT : ( 1 <= k1 & k1 + 1 <= len f ) } by TARSKI:def 4;
consider i1 being Element of NAT such that
A12: Y = LSeg f,i1 and
A13: 1 <= i1 and
A14: i1 + 1 <= len f by A11;
A15: p in LSeg (f /. i1),(f /. (i1 + 1)) by A10, A12, A13, A14, TOPREAL1:def 5;
proj2 .: ((LSeg (G * i,j),(G * i,k)) /\ (L~ f)) = (proj2 | ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) .: ((LSeg (G * i,j),(G * i,k)) /\ (L~ f)) by RELAT_1:162;
then A16: upper_bound (proj2 .: ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) = upper_bound ((proj2 | ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) .: ([#] ((TOP-REAL 2) | ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))))) by PRE_TOPC:def 10
.= N-bound ((LSeg (G * i,j),(G * i,k)) /\ (L~ f)) ;
A17: 1 <= k by A4, MATRIX_1:39;
A18: 1 <= j by A3, MATRIX_1:39;
1 < i1 + 1 by A13, NAT_1:13;
then i1 + 1 in Seg (len f) by A14, FINSEQ_1:3;
then A19: i1 + 1 in dom f by FINSEQ_1:def 3;
then consider io, jo being Element of NAT such that
A20: [io,jo] in Indices G and
A21: f /. (i1 + 1) = G * io,jo by A1, GOBOARD1:def 11;
A22: ( 1 <= io & io <= len G ) by A20, MATRIX_1:39;
A23: 1 <= jo by A20, MATRIX_1:39;
A24: p `2 = (N-min ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) `2 by A7, PSCOMP_1:98
.= upper_bound (proj2 .: ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) by A16, EUCLID:56 ;
A25: ( 1 <= i & i <= len G ) by A3, MATRIX_1:39;
A26: k <= width G by A4, MATRIX_1:39;
then A27: (G * i,j) `2 <= (G * i,k) `2 by A5, A6, A18, SPRECT_3:24;
then A28: (G * i,j) `2 <= p `2 by A9, TOPREAL1:10;
A29: p `2 <= (G * i,k) `2 by A9, A27, TOPREAL1:10;
A30: j <= width G by A3, MATRIX_1:39;
then A31: (G * i,j) `1 = (G * i,1) `1 by A6, A18, GOBOARD5:3
.= (G * i,k) `1 by A25, A17, A26, GOBOARD5:3 ;
A32: jo <= width G by A20, MATRIX_1:39;
i1 <= len f by A14, NAT_1:13;
then i1 in Seg (len f) by A13, FINSEQ_1:3;
then A33: i1 in dom f by FINSEQ_1:def 3;
then consider i0, j0 being Element of NAT such that
A34: [i0,j0] in Indices G and
A35: f /. i1 = G * i0,j0 by A1, GOBOARD1:def 11;
A36: ( 1 <= i0 & i0 <= len G ) by A34, MATRIX_1:39;
A37: 1 <= j0 by A34, MATRIX_1:39;
A38: j0 <= width G by A34, MATRIX_1:39;
A39: f is special by A1, A33, JORDAN8:7, RELAT_1:60;
ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p )

proof

per cases ) `1 = (f /. (i1 + 1)) `1 or (f /. i1) `2 = (f /. (i1 + 1)) `2 ) by A13, A14, A39, TOPREAL1:def 7;

suppose A40: (f /. i1) `1 = (f /. (i1 + 1)) `1 ; :: thesis:

(G * io,j) `1 = (G * io,1) `1 by A18, A30, A22, GOBOARD5:3
.= (G * io,jo) `1 by A22, A23, A32, GOBOARD5:3
.= p `1 by A15, A21, A40, GOBOARD7:5
.= (G * i,j) `1 by A31, A9, GOBOARD7:5 ;
then ( io <= i & io >= i ) by A6, A18, A30, A22, GOBOARD5:4;
then A42: i = io by XXREAL_0:1;
(G * i0,j) `1 = (G * i0,1) `1 by A18, A30, A36, GOBOARD5:3
.= (G * i0,j0) `1 by A36, A37, A38, GOBOARD5:3
.= p `1 by A15, A35, A40, GOBOARD7:5
.= (G * i,j) `1 by A31, A9, GOBOARD7:5 ;
then ( i0 <= i & i0 >= i ) by A6, A18, A30, A36, GOBOARD5:4;
then A44: i = i0 by XXREAL_0:1;
thus ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) :: thesis:

proof

per cases ) `2 <= (f /. (i1 + 1)) `2 or (f /. i1) `2 > (f /. (i1 + 1)) `2 ) ;

suppose A45: (f /. i1) `2 <= (f /. (i1 + 1)) `2 ; :: thesis:

thus ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) :: thesis:

proof

per cases ;

suppose A46: f /. (i1 + 1) in LSeg (G * i,j),(G * i,k) ; :: thesis:

1 + 1 <= i1 + 1 by A13, XREAL_1:8;
then f /. (i1 + 1) in L~ f by A14, A19, GOBOARD1:16, XXREAL_0:2;
then f /. (i1 + 1) in X1 by A46, XBOOLE_0:def 4;
then A47: p `2 >= (f /. (i1 + 1)) `2 by A16, A24, PSCOMP_1:71;
take n = jo; :: thesis:
A48: p in LSeg (G * i,j),(G * i,k) by A8, XBOOLE_0:def 4;
p `2 <= (f /. (i1 + 1)) `2 by A15, A45, TOPREAL1:10;
then p `2 = (f /. (i1 + 1)) `2 by A47, XXREAL_0:1;
then A49: p `2 = (G * 1,jo) `2 by A21, A22, A23, A32, GOBOARD5:2
.= (G * i,n) `2 by A6, A23, A32, GOBOARD5:2 ;
A50: (G * i,j) `2 <= (G * i,k) `2 by A5, A6, A18, A26, SPRECT_3:24;
then (G * i,j) `2 <= (G * i,n) `2 by A48, A49, TOPREAL1:10;
hence j <= n by A6, A30, A23, GOBOARD5:5; :: thesis:
(G * i,n) `2 <= (G * i,k) `2 by A48, A49, A50, TOPREAL1:10;
hence n <= k by A25, A17, A32, GOBOARD5:5; :: thesis:
p `1 = (G * i,j) `1 by A31, A48, GOBOARD7:5
.= (G * i,1) `1 by A6, A18, A30, GOBOARD5:3
.= (G * i,n) `1 by A6, A23, A32, GOBOARD5:3 ;
hence G * i,n = p by A49, TOPREAL3:11; :: thesis:

end;

suppose A51: not f /. (i1 + 1) in LSeg (G * i,j),(G * i,k) ; :: thesis:

A52: (f /. (i1 + 1)) `1 = p `1 by A15, A40, GOBOARD7:5
.= (G * i,j) `1 by A31, A9, GOBOARD7:5 ;
thus ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) :: thesis:

proof

per cases ) `2 > (G * i,k) `2 or (f /. (i1 + 1)) `2 < (G * i,j) `2 ) by A31, A51, A52, GOBOARD7:8;

suppose A53: (f /. (i1 + 1)) `2 > (G * i,k) `2 ; :: thesis:

p `2 >= (G * i0,j0) `2 by A15, A35, A45, TOPREAL1:10;
then p `2 >= (G * 1,j0) `2 by A36, A37, A38, GOBOARD5:2;
then p `2 >= (G * i,j0) `2 by A6, A37, A38, GOBOARD5:2;
then (G * i,k) `2 >= (G * i,j0) `2 by A29, XXREAL_0:2;
then A54: k >= j0 by A25, A17, A38, GOBOARD5:5;
(abs (i0 - io)) + (abs (j0 - jo)) = 1 by A1, A33, A19, A34, A35, A20, A21, GOBOARD1:def 11;
then 0 + (abs (j0 - jo)) = 1 by A44, A42, ABSVALUE:7;
then A55: abs (- (j0 - jo)) = 1 by COMPLEX1:138;
j0 <= jo + 0 by A35, A21, A36, A38, A23, A44, A42, A45, GOBOARD5:5;
then j0 - jo <= 0 by XREAL_1:22;
then jo - j0 = 1 by A55, ABSVALUE:def 1;
then A56: j0 + 1 = jo + 0 ;
( (G * i,jo) `2 > (G * i,k) `2 & jo >= k ) by A21, A25, A26, A23, A42, A53, GOBOARD5:5;
then jo > k by XXREAL_0:1;
then j0 >= k by A56, NAT_1:13;
then A57: k = j0 by A54, XXREAL_0:1;
take n = j0; :: thesis:
A58: p `1 = (G * i,j) `1 by A31, A9, GOBOARD7:5
.= (G * i,1) `1 by A6, A18, A30, GOBOARD5:3
.= (G * i,n) `1 by A6, A37, A38, GOBOARD5:3 ;
p `2 >= (G * i0,j0) `2 by A15, A35, A45, TOPREAL1:10;
then p `2 >= (G * 1,j0) `2 by A36, A37, A38, GOBOARD5:2;
then p `2 >= (G * i,j0) `2 by A6, A37, A38, GOBOARD5:2;
then p `2 = (G * i,k) `2 by A29, A57, XXREAL_0:1;
hence ( j <= n & n <= k & G * i,n = p ) by A5, A57, A58, TOPREAL3:11; :: thesis:

end;

suppose A59: (f /. (i1 + 1)) `2 < (G * i,j) `2 ; :: thesis:

p `2 <= (f /. (i1 + 1)) `2 by A15, A45, TOPREAL1:10;
hence ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) by A28, A59, XXREAL_0:2; :: thesis:

end;

suppose A60: (f /. i1) `2 > (f /. (i1 + 1)) `2 ; :: thesis:

thus ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) :: thesis:

proof

per cases ;

suppose A61: f /. i1 in LSeg (G * i,j),(G * i,k) ; :: thesis:

1 + 1 <= i1 + 1 by A13, XREAL_1:8;
then f /. i1 in L~ f by A14, A33, GOBOARD1:16, XXREAL_0:2;
then f /. i1 in X1 by A61, XBOOLE_0:def 4;
then A62: p `2 >= (f /. i1) `2 by A16, A24, PSCOMP_1:71;
take n = j0; :: thesis:
A63: p in LSeg (G * i,j),(G * i,k) by A8, XBOOLE_0:def 4;
p `2 <= (f /. i1) `2 by A15, A60, TOPREAL1:10;
then p `2 = (f /. i1) `2 by A62, XXREAL_0:1;
then A64: p `2 = (G * 1,j0) `2 by A35, A36, A37, A38, GOBOARD5:2
.= (G * i,n) `2 by A6, A37, A38, GOBOARD5:2 ;
A65: (G * i,j) `2 <= (G * i,k) `2 by A5, A6, A18, A26, SPRECT_3:24;
then (G * i,j) `2 <= (G * i,n) `2 by A63, A64, TOPREAL1:10;
hence j <= n by A6, A30, A37, GOBOARD5:5; :: thesis:
(G * i,n) `2 <= (G * i,k) `2 by A63, A64, A65, TOPREAL1:10;
hence n <= k by A25, A17, A38, GOBOARD5:5; :: thesis:
p `1 = (G * i,j) `1 by A31, A63, GOBOARD7:5
.= (G * i,1) `1 by A6, A18, A30, GOBOARD5:3
.= (G * i,n) `1 by A6, A37, A38, GOBOARD5:3 ;
hence G * i,n = p by A64, TOPREAL3:11; :: thesis:

end;

suppose A66: not f /. i1 in LSeg (G * i,j),(G * i,k) ; :: thesis:

A67: (f /. i1) `1 = p `1 by A15, A40, GOBOARD7:5
.= (G * i,j) `1 by A31, A9, GOBOARD7:5 ;
thus ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) :: thesis:

proof

per cases ) `2 > (G * i,k) `2 or (f /. i1) `2 < (G * i,j) `2 ) by A31, A66, A67, GOBOARD7:8;

suppose A68: (f /. i1) `2 > (G * i,k) `2 ; :: thesis:

p `2 >= (G * io,jo) `2 by A15, A21, A60, TOPREAL1:10;
then p `2 >= (G * 1,jo) `2 by A22, A23, A32, GOBOARD5:2;
then p `2 >= (G * i,jo) `2 by A6, A23, A32, GOBOARD5:2;
then (G * i,k) `2 >= (G * i,jo) `2 by A29, XXREAL_0:2;
then A69: k >= jo by A25, A17, A32, GOBOARD5:5;
jo <= j0 + 0 by A35, A21, A36, A37, A32, A44, A42, A60, GOBOARD5:5;
then jo - j0 <= 0 by XREAL_1:22;
then A70: - (jo - j0) >= - 0 ;
(abs (i0 - io)) + (abs (j0 - jo)) = 1 by A1, A33, A19, A34, A35, A20, A21, GOBOARD1:def 11;
then 0 + (abs (j0 - jo)) = 1 by A44, A42, ABSVALUE:7;
then j0 - jo = 1 by A70, ABSVALUE:def 1;
then A71: jo + 1 = j0 - 0 ;
( (G * i,j0) `2 > (G * i,k) `2 & j0 >= k ) by A35, A25, A26, A37, A44, A68, GOBOARD5:5;
then j0 > k by XXREAL_0:1;
then jo >= k by A71, NAT_1:13;
then A72: k = jo by A69, XXREAL_0:1;
take n = jo; :: thesis:
A73: p `1 = (G * i,j) `1 by A31, A9, GOBOARD7:5
.= (G * i,1) `1 by A6, A18, A30, GOBOARD5:3
.= (G * i,n) `1 by A6, A23, A32, GOBOARD5:3 ;
p `2 >= (G * io,jo) `2 by A15, A21, A60, TOPREAL1:10;
then p `2 >= (G * 1,jo) `2 by A22, A23, A32, GOBOARD5:2;
then p `2 >= (G * i,jo) `2 by A6, A23, A32, GOBOARD5:2;
then p `2 = (G * i,k) `2 by A29, A72, XXREAL_0:1;
hence ( j <= n & n <= k & G * i,n = p ) by A5, A72, A73, TOPREAL3:11; :: thesis:

end;

suppose A74: (f /. i1) `2 < (G * i,j) `2 ; :: thesis:

p `2 <= (f /. i1) `2 by A15, A60, TOPREAL1:10;
hence ex n being Element of NAT st
( j <= n & n <= k & G * i,n = p ) by A28, A74, XXREAL_0:2; :: thesis:

end;

suppose A75: (f /. i1) `2 = (f /. (i1 + 1)) `2 ; :: thesis:

take n = j0; :: thesis:
p `2 = (f /. i1) `2 by A15, A75, GOBOARD7:6;
then A76: p `2 = (G * 1,j0) `2 by A35, A36, A37, A38, GOBOARD5:2
.= (G * i,n) `2 by A6, A37, A38, GOBOARD5:2 ;
A77: (G * i,j) `2 <= (G * i,k) `2 by A5, A6, A18, A26, SPRECT_3:24;
then (G * i,j) `2 <= (G * i,n) `2 by A9, A76, TOPREAL1:10;
hence j <= n by A6, A30, A37, GOBOARD5:5; :: thesis:
(G * i,n) `2 <= (G * i,k) `2 by A9, A76, A77, TOPREAL1:10;
hence n <= k by A25, A17, A38, GOBOARD5:5; :: thesis:
p `1 = (G * i,j) `1 by A31, A9, GOBOARD7:5
.= (G * i,1) `1 by A6, A18, A30, GOBOARD5:3
.= (G * i,n) `1 by A6, A37, A38, GOBOARD5:3 ;
hence G * i,n = p by A76, TOPREAL3:11; :: thesis:

end;

hence ex n being Element of NAT st
( j <= n & n <= k & (G * i,n) `2 = upper_bound (proj2 .: ((LSeg (G * i,j),(G * i,k)) /\ (L~ f))) ) by A24; :: thesis: