let a, b, c, d be Real; :: thesis:
let p1, p2 be Point of (TOP-REAL 2); :: thesis:
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 in LSeg (|[b,c]|,|[a,c]|) and
A4: p1 <> W-min (rectangle (a,b,c,d)) ; :: thesis:
A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50;
A6: p1 `2 = c by A1, A3, Th3;
A7: p1 `1 <= b by A1, A3, Th3;
thus ( LE p1,p2, rectangle (a,b,c,d) implies ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) :: thesis:

proof

assume A8: LE p1,p2, rectangle (a,b,c,d) ; :: thesis:
then A9: p2 in rectangle (a,b,c,d) by A5, 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 A9, XBOOLE_0:def 3;
then A10: ( 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:

per cases by A10, XBOOLE_0:def 3;

case p2 in LSeg (|[a,c]|,|[a,d]|) ; :: thesis:

then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th59;
hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; :: thesis:

end;

case p2 in LSeg (|[a,d]|,|[b,d]|) ; :: thesis:

then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th60;
hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; :: thesis:

end;

case p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis:

then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th61;
hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; :: thesis:

end;

case p2 in LSeg (|[b,c]|,|[a,c]|) ; :: thesis:

hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th58; :: thesis:

end;

hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ; :: thesis:

end;

thus ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) implies LE p1,p2, rectangle (a,b,c,d) ) :: thesis:

proof

assume that
A11: p2 in LSeg (|[b,c]|,|[a,c]|) and
A12: p1 `1 >= p2 `1 and
A13: p2 <> W-min (rectangle (a,b,c,d)) ; :: thesis:

now :: thesis:

per cases by A11, A12;

case A14: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 ) ; :: thesis:

then A15: p2 `2 = c by A1, Th3;
A16: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
then A17: p2 in Lower_Arc (rectangle (a,b,c,d)) by A14, XBOOLE_0:def 3;
A18: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A16, XBOOLE_0:def 3;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d))))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (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) | (Lower_Arc (rectangle (a,b,c,d)))); :: thesis:
let s1, s2 be Real; :: thesis:
assume that
A19: g is being_homeomorphism and
A20: g . 0 = E-max (rectangle (a,b,c,d)) and
g . 1 = W-min (rectangle (a,b,c,d)) and
A21: g . s1 = p1 and
A22: 0 <= s1 and
A23: s1 <= 1 and
A24: g . s2 = p2 and
A25: 0 <= s2 and
A26: s2 <= 1 ; :: thesis:
A27: dom g = the carrier of I[01] by FUNCT_2:def 1;
A28: g is one-to-one by A19, TOPS_2:def 5;
A29: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_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 A19, TOPS_2:def 5;
then A30: 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 ;
A31: 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 A32: ( ( 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 A31, 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
A33: 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
A34: h is continuous by A32, JGRAPH_2:19;
reconsider k = h * g1 as Function of I[01],R^1 ;
A35: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46;

now :: thesis:

proof

let x1, x2 be set ; :: thesis:
assume that
A52: x1 in dom k and
A53: x2 in dom k and
A54: k . x1 = k . x2 ; :: thesis:
reconsider r1 = x1 as Point of I[01] by A52;
reconsider r2 = x2 as Point of I[01] by A53;
A55: k . x1 = h . (g1 . x1) by A52, FUNCT_1:12
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A33
.= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def 5
.= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def 6 ;
A56: k . x2 = h . (g1 . x2) by A53, FUNCT_1:12
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A33
.= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def 5
.= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def 6 ;
A57: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A29;
A58: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A29;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A57;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A58;

now :: thesis:

per cases by A16, A29, XBOOLE_0:def 3;

case A59: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis:

then A60: gr1 `1 = b by A2, Th1;
gr2 `1 = b by A2, A59, Th1;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A54, A55, A56, A60, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A27, A28, FUNCT_1:def 4; :: thesis:

end;

case A61: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis:

then A62: gr1 `1 = b by A2, Th1;
A63: c <= gr1 `2 by A2, A61, Th1;
A64: gr2 `2 = c by A1, A61, Th3;
A65: gr2 `1 <= b by A1, A61, Th3;
A66: b + (gr1 `2) = (gr2 `1) + c by A2, A54, A55, A56, A61, A64, Th1;

A67: now :: thesis:

assume b <> gr2 `1 ; :: thesis:
then b > gr2 `1 by A65, XXREAL_0:1;
hence contradiction by A54, A55, A56, A62, A63, A64, XREAL_1:8; :: thesis:

end;

now :: thesis:

assume gr1 `2 <> c ; :: thesis:
then c < gr1 `2 by A63, XXREAL_0:1;
hence contradiction by A65, A66, XREAL_1:8; :: thesis:

end;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A62, A64, A67, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A27, A28, FUNCT_1:def 4; :: thesis:

end;

case A68: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis:

then A69: gr2 `1 = b by A2, Th1;
A70: c <= gr2 `2 by A2, A68, Th1;
A71: gr1 `2 = c by A1, A68, Th3;
A72: gr1 `1 <= b by A1, A68, Th3;
A73: b + (gr2 `2) = (gr1 `1) + c by A1, A54, A55, A56, A68, A69, Th3;

A74: now :: thesis:

assume b <> gr1 `1 ; :: thesis:
then b > gr1 `1 by A72, XXREAL_0:1;
hence contradiction by A70, A73, XREAL_1:8; :: thesis:

end;

now :: thesis:

assume gr2 `2 <> c ; :: thesis:
then c < gr2 `2 by A70, XXREAL_0:1;
hence contradiction by A54, A55, A56, A69, A71, A72, XREAL_1:8; :: thesis:

end;

then |[(gr2 `1),(gr2 `2)]| = g . r1 by A69, A71, A74, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A27, A28, FUNCT_1:def 4; :: thesis:

end;

case A75: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis:

then A76: gr1 `2 = c by A1, Th3;
gr2 `2 = c by A1, A75, Th3;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A54, A55, A56, A76, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A27, A28, FUNCT_1:def 4; :: thesis:

end;

hence x1 = x2 ; :: thesis:

end;

A77: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
then s1 in dom k by A22, A23, XXREAL_1:1;
then rxc = s1 by A45, A49, A50, A51, A77;
hence contradiction by A36, A49, XXREAL_1:1; :: thesis:

end;

hence s1 <= s2 ; :: thesis:

end;

then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A17, A18, JORDAN5C:def 3;
hence LE p1,p2, rectangle (a,b,c,d) by A13, A17, A18, JORDAN6:def 10; :: thesis:

end;

case A78: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; :: thesis:

then A79: p2 `2 = c by A1, Th3;
A80: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
then A81: p2 in Lower_Arc (rectangle (a,b,c,d)) by A78, XBOOLE_0:def 3;
A82: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A80, XBOOLE_0:def 3;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d))))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (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) | (Lower_Arc (rectangle (a,b,c,d)))); :: thesis:
let s1, s2 be Real; :: thesis:
assume that
A83: g is being_homeomorphism and
A84: g . 0 = E-max (rectangle (a,b,c,d)) and
g . 1 = W-min (rectangle (a,b,c,d)) and
A85: g . s1 = p1 and
A86: 0 <= s1 and
A87: s1 <= 1 and
A88: g . s2 = p2 and
A89: 0 <= s2 and
A90: s2 <= 1 ; :: thesis:
A91: dom g = the carrier of I[01] by FUNCT_2:def 1;
A92: g is one-to-one by A83, TOPS_2:def 5;
A93: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_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 A83, TOPS_2:def 5;
then A94: 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 ;
A95: 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 A96: ( ( 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 A95, 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
A97: 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
A98: h is continuous by A96, JGRAPH_2:19;
reconsider k = h * g1 as Function of I[01],R^1 ;
A99: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46;

now :: thesis:

proof

let x1, x2 be set ; :: thesis:
assume that
A116: x1 in dom k and
A117: x2 in dom k and
A118: k . x1 = k . x2 ; :: thesis:
reconsider r1 = x1 as Point of I[01] by A116;
reconsider r2 = x2 as Point of I[01] by A117;
A119: k . x1 = h . (g1 . x1) by A116, FUNCT_1:12
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A97
.= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def 5
.= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def 6 ;
A120: k . x2 = h . (g1 . x2) by A117, FUNCT_1:12
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A97
.= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def 5
.= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def 6 ;
A121: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A93;
A122: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A93;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A121;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A122;

now :: thesis:

per cases by A80, A93, XBOOLE_0:def 3;

case A123: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis:

then A124: gr1 `1 = b by A2, Th1;
gr2 `1 = b by A2, A123, Th1;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A118, A119, A120, A124, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A91, A92, FUNCT_1:def 4; :: thesis:

end;

case A125: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis:

then A126: gr1 `1 = b by A2, Th1;
A127: c <= gr1 `2 by A2, A125, Th1;
A128: gr2 `2 = c by A1, A125, Th3;
A129: gr2 `1 <= b by A1, A125, Th3;
A130: b + (gr1 `2) = (gr2 `1) + c by A2, A118, A119, A120, A125, A128, Th1;

A131: now :: thesis:

assume b <> gr2 `1 ; :: thesis:
then b > gr2 `1 by A129, XXREAL_0:1;
hence contradiction by A118, A119, A120, A126, A127, A128, XREAL_1:8; :: thesis:

end;

now :: thesis:

assume gr1 `2 <> c ; :: thesis:
then c < gr1 `2 by A127, XXREAL_0:1;
hence contradiction by A129, A130, XREAL_1:8; :: thesis:

end;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A126, A128, A131, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A91, A92, FUNCT_1:def 4; :: thesis:

end;

case A132: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis:

then A133: gr2 `1 = b by A2, Th1;
A134: c <= gr2 `2 by A2, A132, Th1;
A135: gr1 `2 = c by A1, A132, Th3;
A136: gr1 `1 <= b by A1, A132, Th3;
A137: b + (gr2 `2) = (gr1 `1) + c by A1, A118, A119, A120, A132, A133, Th3;

A138: now :: thesis:

assume b <> gr1 `1 ; :: thesis:
then b > gr1 `1 by A136, XXREAL_0:1;
hence contradiction by A134, A137, XREAL_1:8; :: thesis:

end;

now :: thesis:

assume gr2 `2 <> c ; :: thesis:
then c < gr2 `2 by A134, XXREAL_0:1;
hence contradiction by A118, A119, A120, A133, A135, A136, XREAL_1:8; :: thesis:

end;

then |[(gr2 `1),(gr2 `2)]| = g . r1 by A133, A135, A138, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A91, A92, FUNCT_1:def 4; :: thesis:

end;

case A139: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis:

then A140: gr1 `2 = c by A1, Th3;
gr2 `2 = c by A1, A139, Th3;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A118, A119, A120, A140, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A91, A92, FUNCT_1:def 4; :: thesis:

end;

hence x1 = x2 ; :: thesis:

end;

A141: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
then s1 in dom k by A86, A87, XXREAL_1:1;
then rxc = s1 by A109, A113, A114, A115, A141;
hence contradiction by A100, A113, XXREAL_1:1; :: thesis:

end;

hence s1 <= s2 ; :: thesis:

end;

then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A81, A82, JORDAN5C:def 3;
hence LE p1,p2, rectangle (a,b,c,d) by A78, A81, A82, JORDAN6:def 10; :: thesis:

end;

hence LE p1,p2, rectangle (a,b,c,d) ; :: thesis:

end;