let a, b, c, d be real number ; :: thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg |[a,c]|,|[a,d]| holds
( LE p1,p2, rectangle a,b,c,d iff ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| implies ( LE p1,p2, rectangle a,b,c,d iff ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| ; :: thesis: ( LE p1,p2, rectangle a,b,c,d iff ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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, Th60;
Upper_Arc (rectangle a,b,c,d) = (LSeg |[a,c]|,|[a,d]|) \/ (LSeg |[a,d]|,|[b,d]|) by A1, A2, Th61;
then A5: LSeg |[a,c]|,|[a,d]| c= Upper_Arc (rectangle a,b,c,d) by XBOOLE_1:7;
A6: p1 `1 = a by A2, A3, Th9;
A7: c <= p1 `2 by A2, A3, Th9;
A8: p1 `2 <= d by A2, A3, Th9;
thus ( not LE p1,p2, rectangle a,b,c,d or ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A9: LE p1,p2, rectangle a,b,c,d ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A10: p1 in rectangle a,b,c,d by A4, JORDAN7:5;
A11: p2 in rectangle a,b,c,d by A4, A9, 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 A11, XBOOLE_0:def 3;
then A12: ( 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
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 A12, XBOOLE_0:def 3;
case p2 in LSeg |[a,c]|,|[a,d]| ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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, A9, Th65; :: thesis: verum
end;
case p2 in LSeg |[a,d]|,|[b,d]| ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 p2 in LSeg |[b,d]|,|[b,c]| ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A13: p2 in LSeg |[b,c]|,|[a,c]| ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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
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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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, A10, JORDAN7:3;
hence ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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, A9, Th60, JORDAN6:72; :: thesis: verum
end;
case p2 <> W-min (rectangle a,b,c,d) ; :: thesis: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A13; :: thesis: verum
end;
end;
end;
hence ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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;
A14: W-min (rectangle a,b,c,d) = |[a,c]| by A1, A2, Th56;
thus ( ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A15: ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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
per cases ( ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) or p2 in LSeg |[a,d]|,|[b,d]| 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 A15;
case ( p2 in LSeg |[a,c]|,|[a,d]| & p1 `2 <= p2 `2 ) ; :: thesis: LE p1,p2, rectangle a,b,c,d
hence LE p1,p2, rectangle a,b,c,d by A1, A2, A3, Th65; :: thesis: verum
end;
case A16: p2 in LSeg |[a,d]|,|[b,d]| ; :: thesis: LE p1,p2, rectangle a,b,c,d
then A17: p2 `2 = d by A1, Th11;
A18: a <= p2 `1 by A1, A16, Th11;
A19: Upper_Arc (rectangle a,b,c,d) = (LSeg |[a,c]|,|[a,d]|) \/ (LSeg |[a,d]|,|[b,d]|) by A1, A2, Th61;
then A20: p2 in Upper_Arc (rectangle a,b,c,d) by A16, XBOOLE_0:def 3;
A21: p1 in Upper_Arc (rectangle a,b,c,d) by A3, A19, 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
A22: g is being_homeomorphism and
A23: g . 0 = W-min (rectangle a,b,c,d) and
g . 1 = E-max (rectangle a,b,c,d) and
A24: g . s1 = p1 and
A25: 0 <= s1 and
A26: s1 <= 1 and
A27: g . s2 = p2 and
A28: 0 <= s2 and
A29: s2 <= 1 ; :: thesis: s1 <= s2
A30: dom g = the carrier of I[01] by FUNCT_2:def 1;
A31: g is one-to-one by A22, TOPS_2:def 5;
A32: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle a,b,c,d))) = Upper_Arc (rectangle a,b,c,d) by PRE_TOPC:29;
then reconsider g1 = g as Function of I[01] ,(TOP-REAL 2) by FUNCT_2:9;
g is continuous by A22, TOPS_2:def 5;
then A33: g1 is continuous by PRE_TOPC:56;
reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:24;
reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:24;
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 ;
A34: 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:66
.= (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_2:39;
then A35: h1 is continuous by PRE_TOPC:62;
( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A34, JGRAPH_2:40;
then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:62;
then consider h being Function of (TOP-REAL 2),R^1 such that
A36: for p being Point of (TOP-REAL 2)
for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds
h . p = r1 + r2 and
A37: h is continuous by A35, JGRAPH_2:29;
reconsider k = h * g1 as Function of I[01] ,R^1 ;
A38: W-min (rectangle a,b,c,d) = |[a,c]| by A1, A2, Th56;
now
assume A39: s1 > s2 ; :: thesis: contradiction
A40: dom g = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
0 in [.0 ,1.] by XXREAL_1:1;
then A41: k . 0 = h . (W-min (rectangle a,b,c,d)) by A23, A40, FUNCT_1:23
.= (h1 . (W-min (rectangle a,b,c,d))) + (h2 . (W-min (rectangle a,b,c,d))) by A36
.= ((W-min (rectangle a,b,c,d)) `1 ) + (proj2 . (W-min (rectangle a,b,c,d))) by PSCOMP_1:def 28
.= ((W-min (rectangle a,b,c,d)) `1 ) + ((W-min (rectangle a,b,c,d)) `2 ) by PSCOMP_1:def 29
.= ((W-min (rectangle a,b,c,d)) `1 ) + c by A38, EUCLID:56
.= a + c by A38, EUCLID:56 ;
s1 in [.0 ,1.] by A25, A26, XXREAL_1:1;
then A42: k . s1 = h . p1 by A24, A40, FUNCT_1:23
.= (proj1 . p1) + (proj2 . p1) by A36
.= (p1 `1 ) + (proj2 . p1) by PSCOMP_1:def 28
.= a + (p1 `2 ) by A6, PSCOMP_1:def 29 ;
A43: s2 in [.0 ,1.] by A28, A29, XXREAL_1:1;
then A44: k . s2 = h . p2 by A27, A40, FUNCT_1:23
.= (proj1 . p2) + (proj2 . p2) by A36
.= (p2 `1 ) + (proj2 . p2) by PSCOMP_1:def 28
.= (p2 `1 ) + d by A17, PSCOMP_1:def 29 ;
A45: k . 0 <= k . s1 by A7, A41, A42, XREAL_1:9;
A46: k . s1 <= k . s2 by A8, A18, A42, A44, XREAL_1:9;
A47: 0 in [.0 ,1.] by XXREAL_1:1;
then A48: [.0 ,s2.] c= [.0 ,1.] by A43, XXREAL_2:def 12;
reconsider B = [.0 ,s2.] as Subset of I[01] by A43, A47, BORSUK_1:83, XXREAL_2:def 12;
A49: B is connected by A28, A43, A47, BORSUK_1:83, BORSUK_4:49;
A50: 0 in B by A28, XXREAL_1:1;
A51: s2 in B by A28, XXREAL_1:1;
A52: k . 0 is Real by XREAL_0:def 1;
A53: k . s2 is Real by XREAL_0:def 1;
k . s1 is Real by XREAL_0:def 1;
then consider xc being Point of I[01] such that
A54: xc in B and
A55: k . xc = k . s1 by A33, A37, A45, A46, A49, A50, A51, A52, A53, TOPREAL5:11;
xc in [.0 ,1.] by BORSUK_1:83;
then reconsider rxc = xc as Real ;
A56: 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
A57: x1 in dom k and
A58: x2 in dom k and
A59: k . x1 = k . x2 ; :: thesis: x1 = x2
reconsider r1 = x1 as Point of I[01] by A57;
reconsider r2 = x2 as Point of I[01] by A58;
A60: k . x1 = h . (g1 . x1) by A57, FUNCT_1:22
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A36
.= ((g1 . r1) `1 ) + (proj2 . (g1 . r1)) by PSCOMP_1:def 28
.= ((g1 . r1) `1 ) + ((g1 . r1) `2 ) by PSCOMP_1:def 29 ;
A61: k . x2 = h . (g1 . x2) by A58, FUNCT_1:22
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A36
.= ((g1 . r2) `1 ) + (proj2 . (g1 . r2)) by PSCOMP_1:def 28
.= ((g1 . r2) `1 ) + ((g1 . r2) `2 ) by PSCOMP_1:def 29 ;
A62: g . r1 in Upper_Arc (rectangle a,b,c,d) by A32;
A63: g . r2 in Upper_Arc (rectangle a,b,c,d) by A32;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A62;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A63;
now
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 A19, A32, XBOOLE_0:def 3;
case A64: ( g . r1 in LSeg |[a,c]|,|[a,d]| & g . r2 in LSeg |[a,c]|,|[a,d]| ) ; :: thesis: x1 = x2
then A65: gr1 `1 = a by A2, Th9;
gr2 `1 = a by A2, A64, Th9;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A59, A60, A61, A65, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A30, A31, FUNCT_1:def 8; :: thesis: verum
end;
case A66: ( g . r1 in LSeg |[a,c]|,|[a,d]| & g . r2 in LSeg |[a,d]|,|[b,d]| ) ; :: thesis: x1 = x2
then A67: gr1 `1 = a by A2, Th9;
A68: gr1 `2 <= d by A2, A66, Th9;
A69: gr2 `2 = d by A1, A66, Th11;
A70: a <= gr2 `1 by A1, A66, Th11;
A71: a + (gr1 `2 ) = (gr2 `1 ) + d by A1, A59, A60, A61, A66, A67, Th11;
A72: now
assume a <> gr2 `1 ; :: thesis: contradiction
then a < gr2 `1 by A70, XXREAL_0:1;
hence contradiction by A68, A71, XREAL_1:10; :: thesis: verum
end;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A67, A69, A72, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A30, A31, FUNCT_1:def 8; :: thesis: verum
end;
case A73: ( g . r1 in LSeg |[a,d]|,|[b,d]| & g . r2 in LSeg |[a,c]|,|[a,d]| ) ; :: thesis: x1 = x2
then A74: gr2 `1 = a by A2, Th9;
A75: gr2 `2 <= d by A2, A73, Th9;
A76: gr1 `2 = d by A1, A73, Th11;
A77: a <= gr1 `1 by A1, A73, Th11;
A78: a + (gr2 `2 ) = (gr1 `1 ) + d by A1, A59, A60, A61, A73, A74, Th11;
A79: now
assume a <> gr1 `1 ; :: thesis: contradiction
then a < gr1 `1 by A77, XXREAL_0:1;
hence contradiction by A75, A78, XREAL_1:10; :: thesis: verum
end;
then |[(gr2 `1 ),(gr2 `2 )]| = g . r1 by A74, A76, A79, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A30, A31, FUNCT_1:def 8; :: thesis: verum
end;
case A80: ( g . r1 in LSeg |[a,d]|,|[b,d]| & g . r2 in LSeg |[a,d]|,|[b,d]| ) ; :: thesis: x1 = x2
then A81: gr1 `2 = d by A1, Th11;
gr2 `2 = d by A1, A80, Th11;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A59, A60, A61, A81, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A30, A31, FUNCT_1:def 8; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
A82: dom k = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
then s1 in dom k by A25, A26, XXREAL_1:1;
then rxc = s1 by A48, A54, A55, A56, A82;
hence contradiction by A39, A54, 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 A20, A21, JORDAN5C:def 3;
hence LE p1,p2, rectangle a,b,c,d by A20, A21, JORDAN6:def 10; :: thesis: verum
end;
case A86: ( 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, Th62;
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, A86, JORDAN6:def 10; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle a,b,c,d ; :: thesis: verum
end;