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 (|[b,d]|,|[b,c]|) holds
( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) ; :: thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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;
A5: p1 `1 = b by A2, A3, Th9;
A6: c <= p1 `2 by A2, A3, Th9;
A7: p1 `2 <= d by A2, A3, Th9;
thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) :: thesis: ( ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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
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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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, Th69;
hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th60, JORDAN6:57; :: thesis: verum
end;
case p2 in LSeg (|[a,d]|,|[b,d]|) ; :: thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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, Th70;
hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th60, JORDAN6:57; :: thesis: verum
end;
case p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th67; :: thesis: verum
end;
case A12: p2 in LSeg (|[b,c]|,|[a,c]|) ; :: thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th60, JORDAN6:57; :: thesis: verum
end;
case p2 <> W-min (rectangle (a,b,c,d)) ; :: thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )
hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; :: thesis: verum
end;
thus ( ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 A13: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) 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 (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A13;
case A14: ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A15: p2 `1 = b by A2, Th9;
W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th56;
then A16: p2 <> W-min (rectangle (a,b,c,d)) by A1, A15, EUCLID:52;
A17: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th62;
then A18: p2 in Lower_Arc (rectangle (a,b,c,d)) by A14, XBOOLE_0:def 3;
A19: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A17, 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: 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
let s1, s2 be Real; :: thesis: ( 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 implies s1 <= s2 )
assume that
A20: g is being_homeomorphism and
A21: g . 0 = E-max (rectangle (a,b,c,d)) and
g . 1 = W-min (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) | (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 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 number 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: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th56;
now
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 . (E-max (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13
.= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A34
.= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def 5
.= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def 6
.= ((E-max (rectangle (a,b,c,d))) `1) + d by A36, EUCLID:52
.= b + d 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
.= b + (p1 `2) by A5, 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
.= b + (p2 `2) by A15, PSCOMP_1:def 6 ;
A43: k . 0 >= k . s1 by A7, A39, A40, XREAL_1:7;
A44: k . s1 >= k . s2 by A14, 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;
A50: k . 0 is Real by XREAL_0:def 1;
A51: 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
A52: xc in B and
A53: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, A50, A51, TOPREAL5:5;
xc in [.0,1.] by BORSUK_1:40;
then reconsider rxc = xc as Real ;
A54: 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
A55: x1 in dom k and
A56: x2 in dom k and
A57: k . x1 = k . x2 ; :: thesis: x1 = x2
reconsider r1 = x1 as Point of I[01] by A55;
reconsider r2 = x2 as Point of I[01] by A56;
A58: k . x1 = h . (g1 . x1) by A55, 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 ;
A59: k . x2 = h . (g1 . x2) by A56, 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 ;
A60: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A30;
A61: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A30;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A60;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A61;
now
per cases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A17, A30, XBOOLE_0:def 3;
case A62: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A63: gr1 `1 = b by A2, Th9;
gr2 `1 = b by A2, A62, Th9;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A63, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A64: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A65: gr1 `1 = b by A2, Th9;
A66: c <= gr1 `2 by A2, A64, Th9;
A67: gr2 `2 = c by A1, A64, Th11;
A68: gr2 `1 <= b by A1, A64, Th11;
A69: b + (gr1 `2) = (gr2 `1) + c by A2, A57, A58, A59, A64, A67, Th9;
A70: now
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by A68, XXREAL_0:1;
hence contradiction by A57, A58, A59, A65, A66, A67, XREAL_1:8; :: thesis: verum
end;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A65, A67, A70, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A71: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A72: gr2 `1 = b by A2, Th9;
A73: c <= gr2 `2 by A2, A71, Th9;
A74: gr1 `2 = c by A1, A71, Th11;
A75: gr1 `1 <= b by A1, A71, Th11;
A76: b + (gr2 `2) = (gr1 `1) + c by A1, A57, A58, A59, A71, A72, Th11;
A77: now
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by A75, XXREAL_0:1;
hence contradiction by A73, A76, XREAL_1:8; :: thesis: verum
end;
then |[(gr2 `1),(gr2 `2)]| = g . r1 by A72, A74, A77, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum
end;
case A78: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A79: gr1 `2 = c by A1, Th11;
gr2 `2 = c by A1, A78, Th11;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A79, 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;
A80: 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, A52, A53, A54, A80;
hence contradiction by A37, A52, XXREAL_1:1; :: thesis: verum
end;
hence s1 <= s2 ; :: thesis: verum
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 A18, A19, JORDAN5C:def 3;
hence LE p1,p2, rectangle (a,b,c,d) by A16, A18, A19, JORDAN6:def 10; :: thesis: verum
end;
case A81: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A82: p2 `2 = c by A1, Th11;
A83: p2 `1 <= b by A1, A81, Th11;
A84: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th62;
then A85: p2 in Lower_Arc (rectangle (a,b,c,d)) by A81, XBOOLE_0:def 3;
A86: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A84, 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: 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
let s1, s2 be Real; :: thesis: ( 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 implies s1 <= s2 )
assume that
A87: g is being_homeomorphism and
A88: g . 0 = E-max (rectangle (a,b,c,d)) and
g . 1 = W-min (rectangle (a,b,c,d)) and
A89: g . s1 = p1 and
A90: 0 <= s1 and
A91: s1 <= 1 and
A92: g . s2 = p2 and
A93: 0 <= s2 and
A94: s2 <= 1 ; :: thesis: s1 <= s2
A95: dom g = the carrier of I[01] by FUNCT_2:def 1;
A96: g is one-to-one by A87, TOPS_2:def 5;
A97: 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 A87, TOPS_2:def 5;
then A98: 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 ;
A99: 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 A100: ( ( 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 A99, 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
A101: 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
A102: h is continuous by A100, JGRAPH_2:19;
reconsider k = h * g1 as Function of I[01],R^1 ;
A103: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th56;
now
assume A104: s1 > s2 ; :: thesis: contradiction
A105: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then A106: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A88, A105, FUNCT_1:13
.= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A101
.= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def 5
.= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def 6
.= ((E-max (rectangle (a,b,c,d))) `1) + d by A103, EUCLID:52
.= b + d by A103, EUCLID:52 ;
s1 in [.0,1.] by A90, A91, XXREAL_1:1;
then A107: k . s1 = h . p1 by A89, A105, FUNCT_1:13
.= (proj1 . p1) + (proj2 . p1) by A101
.= (p1 `1) + (proj2 . p1) by PSCOMP_1:def 5
.= b + (p1 `2) by A5, PSCOMP_1:def 6 ;
A108: s2 in [.0,1.] by A93, A94, XXREAL_1:1;
then A109: k . s2 = h . p2 by A92, A105, FUNCT_1:13
.= (proj1 . p2) + (proj2 . p2) by A101
.= (p2 `1) + (proj2 . p2) by PSCOMP_1:def 5
.= (p2 `1) + c by A82, PSCOMP_1:def 6 ;
A110: k . 0 >= k . s1 by A7, A106, A107, XREAL_1:7;
A111: k . s1 >= k . s2 by A6, A83, A107, A109, XREAL_1:7;
A112: 0 in [.0,1.] by XXREAL_1:1;
then A113: [.0,s2.] c= [.0,1.] by A108, XXREAL_2:def 12;
reconsider B = [.0,s2.] as Subset of I[01] by A108, A112, BORSUK_1:40, XXREAL_2:def 12;
A114: B is connected by A93, A108, A112, BORSUK_1:40, BORSUK_4:24;
A115: 0 in B by A93, XXREAL_1:1;
A116: s2 in B by A93, XXREAL_1:1;
A117: k . 0 is Real by XREAL_0:def 1;
A118: 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
A119: xc in B and
A120: k . xc = k . s1 by A98, A102, A110, A111, A114, A115, A116, A117, A118, TOPREAL5:5;
xc in [.0,1.] by BORSUK_1:40;
then reconsider rxc = xc as Real ;
A121: 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
A122: x1 in dom k and
A123: x2 in dom k and
A124: k . x1 = k . x2 ; :: thesis: x1 = x2
reconsider r1 = x1 as Point of I[01] by A122;
reconsider r2 = x2 as Point of I[01] by A123;
A125: k . x1 = h . (g1 . x1) by A122, FUNCT_1:12
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A101
.= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def 5
.= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def 6 ;
A126: k . x2 = h . (g1 . x2) by A123, FUNCT_1:12
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A101
.= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def 5
.= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def 6 ;
A127: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A97;
A128: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A97;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A127;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A128;
now
per cases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A84, A97, XBOOLE_0:def 3;
case A129: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A130: gr1 `1 = b by A2, Th9;
gr2 `1 = b by A2, A129, Th9;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A124, A125, A126, A130, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A95, A96, FUNCT_1:def 4; :: thesis: verum
end;
case A131: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A132: gr1 `1 = b by A2, Th9;
A133: c <= gr1 `2 by A2, A131, Th9;
A134: gr2 `2 = c by A1, A131, Th11;
A135: gr2 `1 <= b by A1, A131, Th11;
A136: b + (gr1 `2) = (gr2 `1) + c by A2, A124, A125, A126, A131, A134, Th9;
A137: now
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by A135, XXREAL_0:1;
hence contradiction by A124, A125, A126, A132, A133, A134, XREAL_1:8; :: thesis: verum
end;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A132, A134, A137, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A95, A96, FUNCT_1:def 4; :: thesis: verum
end;
case A138: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A139: gr2 `1 = b by A2, Th9;
A140: c <= gr2 `2 by A2, A138, Th9;
A141: gr1 `2 = c by A1, A138, Th11;
A142: gr1 `1 <= b by A1, A138, Th11;
A143: b + (gr2 `2) = (gr1 `1) + c by A1, A124, A125, A126, A138, A139, Th11;
A144: now
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by A142, XXREAL_0:1;
hence contradiction by A140, A143, XREAL_1:8; :: thesis: verum
end;
then |[(gr2 `1),(gr2 `2)]| = g . r1 by A139, A141, A144, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A95, A96, FUNCT_1:def 4; :: thesis: verum
end;
case A145: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A146: gr1 `2 = c by A1, Th11;
gr2 `2 = c by A1, A145, Th11;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by A124, A125, A126, A146, EUCLID:53;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by A95, A96, FUNCT_1:def 4; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
A147: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
then s1 in dom k by A90, A91, XXREAL_1:1;
then rxc = s1 by A113, A119, A120, A121, A147;
hence contradiction by A104, A119, XXREAL_1:1; :: thesis: verum
end;
hence s1 <= s2 ; :: thesis: verum
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 A85, A86, JORDAN5C:def 3;
hence LE p1,p2, rectangle (a,b,c,d) by A81, A85, A86, JORDAN6:def 10; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle (a,b,c,d) ; :: thesis: verum
end;