let a, b, c, d be Real; :: thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) holds
( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) )
let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 in LSeg (|[a,d]|,|[b,d]|) ; :: thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) )
A4: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50;
Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;
then A5: LSeg (|[a,d]|,|[b,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;
A6: p1 `2 = d by A1, A3, Th3;
A7: a <= p1 `1 by A1, A3, Th3;
thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) :: thesis: ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) )
proof
assume A8: LE p1,p2, rectangle (a,b,c,d) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
then A9: p1 in rectangle (a,b,c,d) by A4, JORDAN7:5;
A10: p2 in rectangle (a,b,c,d) by A4, A8, JORDAN7:5;
rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def 3
.= (((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|))) \/ (LSeg (|[b,c]|,|[a,c]|)) by XBOOLE_1:4 ;
then ( p2 in ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|)) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A10, XBOOLE_0:def 3;
then A11: ( p2 in (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by XBOOLE_0:def 3;
now :: thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) )
per cases ( p2 in LSeg (|[a,c]|,|[a,d]|) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A11, XBOOLE_0:def 3;
case p2 in LSeg (|[a,c]|,|[a,d]|) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, Th59;
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; :: thesis: verum
end;
case p2 in LSeg (|[a,d]|,|[b,d]|) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th56; :: thesis: verum
end;
case p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; :: thesis: verum
end;
case A12: p2 in LSeg (|[b,c]|,|[a,c]|) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
now :: thesis: ( ( p2 = W-min (rectangle (a,b,c,d)) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) or ( p2 <> W-min (rectangle (a,b,c,d)) & ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) )
per cases ( p2 = W-min (rectangle (a,b,c,d)) or p2 <> W-min (rectangle (a,b,c,d)) ) ;
case p2 = W-min (rectangle (a,b,c,d)) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
then LE p2,p1, rectangle (a,b,c,d) by A4, A9, JORDAN7:3;
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; :: thesis: verum
end;
case p2 <> W-min (rectangle (a,b,c,d)) ; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A12; :: thesis: verum
end;
end;
end;
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; :: thesis: verum
end;
end;
end;
hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; :: thesis: verum
end;
A13: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46;
thus ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) :: thesis: verum
proof
assume A14: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
now :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 & LE p1,p2, rectangle (a,b,c,d) ) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & LE p1,p2, rectangle (a,b,c,d) ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) & LE p1,p2, rectangle (a,b,c,d) ) )
per cases ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A14;
case A15: ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A16: p2 `2 = d by A1, Th3;
A17: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;
then A18: p2 in Upper_Arc (rectangle (a,b,c,d)) by A15, XBOOLE_0:def 3;
A19: p1 in Upper_Arc (rectangle (a,b,c,d)) by A3, A17, XBOOLE_0:def 3;
for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d))))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))); :: thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A20: g is being_homeomorphism and
A21: g . 0 = W-min (rectangle (a,b,c,d)) and
g . 1 = E-max (rectangle (a,b,c,d)) and
A22: g . s1 = p1 and
A23: 0 <= s1 and
A24: s1 <= 1 and
A25: g . s2 = p2 and
A26: 0 <= s2 and
A27: s2 <= 1 ; :: thesis: s1 <= s2
A28: dom g = the carrier of I[01] by FUNCT_2:def 1;
A29: g is one-to-one by A20, TOPS_2:def 5;
A30: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8;
then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7;
g is continuous by A20, TOPS_2:def 5;
then A31: g1 is continuous by PRE_TOPC:26;
reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ;
reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ;
A32: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3
.= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36
.= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ;
then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29;
then A33: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32;
( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A32, JGRAPH_2:30;
then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32;
then consider h being Function of (TOP-REAL 2),R^1 such that
A34: for p being Point of (TOP-REAL 2)
for r1, r2 being Real st h1 . p = r1 & h2 . p = r2 holds
h . p = r1 + r2 and
A35: h is continuous by A33, JGRAPH_2:19;
reconsider k = h * g1 as Function of I[01],R^1 ;
A36: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46;
now :: thesis: not s1 > s2
assume A37: s1 > s2 ; :: thesis: contradiction
A38: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then A39: k . 0 = h . (W-min (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13
.= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A34
.= ((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d)))) by PSCOMP_1:def 5
.= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def 6
.= ((W-min (rectangle (a,b,c,d))) `1) + c by A36, EUCLID:52
.= a + c by A36, EUCLID:52 ;
s1 in [.0,1.] by A23, A24, XXREAL_1:1;
then A40: k . s1 = h . p1 by A22, A38, FUNCT_1:13
.= (proj1 . p1) + (proj2 . p1) by A34
.= (p1 `1) + (proj2 . p1) by PSCOMP_1:def 5
.= (p1 `1) + d by A6, PSCOMP_1:def 6 ;
A41: s2 in [.0,1.] by A26, A27, XXREAL_1:1;
then A42: k . s2 = h . p2 by A25, A38, FUNCT_1:13
.= (proj1 . p2) + (proj2 . p2) by A34
.= (p2 `1) + (proj2 . p2) by PSCOMP_1:def 5
.= (p2 `1) + d by A16, PSCOMP_1:def 6 ;
A43: k . 0 <= k . s1 by A2, A7, A39, A40, XREAL_1:7;
A44: k . s1 <= k . s2 by A15, A40, A42, XREAL_1:7;
A45: 0 in [.0,1.] by XXREAL_1:1;
then A46: [.0,s2.] c= [.0,1.] by A41, XXREAL_2:def 12;
reconsider B = [.0,s2.] as Subset of I[01] by A41, A45, BORSUK_1:40, XXREAL_2:def 12;
A47: B is connected by A26, A41, A45, BORSUK_1:40, BORSUK_4:24;
A48: 0 in B by A26, XXREAL_1:1;
A49: s2 in B by A26, XXREAL_1:1;
consider xc being Point of I[01] such that
A50: xc in B and
A51: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, TOPREAL5:5;
reconsider rxc = xc as Real ;
A52: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds
x1 = x2
proof
let x1, x2 be set ; :: thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 )
assume that
A53: x1 in dom k and
A54: x2 in dom k and
A55: k . x1 = k . x2 ; :: thesis: x1 = x2
reconsider r1 = x1 as Point of I[01] by A53;
reconsider r2 = x2 as Point of I[01] by A54;
A56: k . x1 = h . (g1 . x1) by A53, FUNCT_1:12
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A34
.= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def 5
.= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def 6 ;
A57: k . x2 = h . (g1 . x2) by A54, FUNCT_1:12
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A34
.= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def 5
.= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def 6 ;
A58: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A30;
A59: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A30;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A58;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A59;
now :: thesis: ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) )
per cases ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ) by A17, A30, XBOOLE_0:def 3;
case A60: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; :: thesis: x1 = x2
then A61: gr1 `1 = a by A2, Th1;
gr2 `1 = a by A2, A60, Th1;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A61, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A62: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; :: thesis: x1 = x2
then A63: gr1 `1 = a by A2, Th1;
A64: gr1 `2 <= d by A2, A62, Th1;
A65: gr2 `2 = d by A1, A62, Th3;
A66: a <= gr2 `1 by A1, A62, Th3;
A67: a + (gr1 `2) = (gr2 `1) + d by A1, A55, A56, A57, A62, A63, Th3;
A68: now :: thesis: not a <> gr2 `1
assume a <> gr2 `1 ; :: thesis: contradiction
then a < gr2 `1 by A66, XXREAL_0:1;
hence contradiction by A64, A67, XREAL_1:8; :: thesis: verum
end;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A63, A65, A68, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A69: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; :: thesis: x1 = x2
then A70: gr2 `1 = a by A2, Th1;
A71: gr2 `2 <= d by A2, A69, Th1;
A72: gr1 `2 = d by A1, A69, Th3;
A73: a <= gr1 `1 by A1, A69, Th3;
A74: a + (gr2 `2) = (gr1 `1) + d by A1, A55, A56, A57, A69, A70, Th3;
A75: now :: thesis: not a <> gr1 `1
assume a <> gr1 `1 ; :: thesis: contradiction
then a < gr1 `1 by A73, XXREAL_0:1;
hence contradiction by A71, A74, XREAL_1:8; :: thesis: verum
end;
then |[(gr2 `1),(gr2 `2)]| = g . r1 by A70, A72, A75, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A76: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; :: thesis: x1 = x2
then A77: gr1 `2 = d by A1, Th3;
gr2 `2 = d by A1, A76, Th3;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A77, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
A78: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
then s1 in dom k by A23, A24, XXREAL_1:1;
then rxc = s1 by A46, A50, A51, A52, A78;
hence contradiction by A37, A50, XXREAL_1:1; :: thesis: verum
end;
hence s1 <= s2 ; :: thesis: verum
end;
then LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A18, A19, JORDAN5C:def 3;
hence LE p1,p2, rectangle (a,b,c,d) by A18, A19, JORDAN6:def 10; :: thesis: verum
end;
case A79: p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A80: p2 `1 = b by TOPREAL3:11;
Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
then A81: LSeg (|[b,d]|,|[b,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;
p2 <> W-min (rectangle (a,b,c,d)) by A1, A13, A80, EUCLID:52;
hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A79, A81, JORDAN6:def 10; :: thesis: verum
end;
case A82: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
then LSeg (|[b,c]|,|[a,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;
hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A82, JORDAN6:def 10; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle (a,b,c,d) ; :: thesis: verum
end;