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 A1: ( a < b & c < d & 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) ) ) )
then A2: rectangle a,b,c,d is being_simple_closed_curve by Th60;
A3: ( p1 `1 = b & c <= p1 `2 & p1 `2 <= d ) by A1, 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 A4: 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 A5: ( p1 in rectangle a,b,c,d & p2 in rectangle a,b,c,d ) by A2, 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 A5, XBOOLE_0:def 3;
then A6: ( 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 A6, 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, 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, A4, Th60, JORDAN6:72; :: 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, 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, A4, Th60, JORDAN6:72; :: 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, A4, Th67; :: thesis: verum
end;
case A7: 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 A2, A5, 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, A4, Th60, JORDAN6:72; :: 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 A7; :: 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 A8: ( ( 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 A8;
case A9: ( p2 in LSeg |[b,d]|,|[b,c]| & p1 `2 >= p2 `2 ) ; :: thesis: LE p1,p2, rectangle a,b,c,d
then A10: ( p2 `1 = b & c <= p2 `2 & p2 `2 <= d ) by A1, Th9;
W-min (rectangle a,b,c,d) = |[a,c]| by A1, Th56;
then A11: p2 <> W-min (rectangle a,b,c,d) by A1, A10, EUCLID:56;
A12: Lower_Arc (rectangle a,b,c,d) = (LSeg |[b,d]|,|[b,c]|) \/ (LSeg |[b,c]|,|[a,c]|) by A1, Th62;
then A13: p2 in Lower_Arc (rectangle a,b,c,d) by A9, XBOOLE_0:def 3;
A14: p1 in Lower_Arc (rectangle a,b,c,d) by A1, A12, XBOOLE_0:def 3;
LE p1,p2, Lower_Arc (rectangle a,b,c,d), E-max (rectangle a,b,c,d), W-min (rectangle a,b,c,d)
proof
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 A15: ( 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 ) ; :: thesis: s1 <= s2
A16: dom g = the carrier of I[01] by FUNCT_2:def 1;
A17: g is one-to-one by A15, TOPS_2:def 5;
A18: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) = Lower_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 A15, TOPS_2:def 5;
then A19: 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 ;
A29: 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 B24: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:39;
A21: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by B24, PRE_TOPC:62;
BB: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A29, JGRAPH_2:40;
( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by BB, PRE_TOPC:62;
then consider h being Function of (TOP-REAL 2),R^1 such that
A22: ( ( 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 ) & h is continuous ) by A21, JGRAPH_2:29;
reconsider k = h * g1 as Function of I[01] ,R^1 ;
A23: k is continuous Function of I[01] ,R^1 by A19, A22;
A24: E-max (rectangle a,b,c,d) = |[b,d]| by A1, Th56;
now
assume A25: s1 > s2 ; :: thesis: contradiction
A26: dom g = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
0 in [.0 ,1.] by XXREAL_1:1;
then A27: k . 0 = h . (E-max (rectangle a,b,c,d)) by A15, A26, FUNCT_1:23
.= (h1 . (E-max (rectangle a,b,c,d))) + (h2 . (E-max (rectangle a,b,c,d))) by A22
.= ((E-max (rectangle a,b,c,d)) `1 ) + (proj2 . (E-max (rectangle a,b,c,d))) by PSCOMP_1:def 28
.= ((E-max (rectangle a,b,c,d)) `1 ) + ((E-max (rectangle a,b,c,d)) `2 ) by PSCOMP_1:def 29
.= ((E-max (rectangle a,b,c,d)) `1 ) + d by A24, EUCLID:56
.= b + d by A24, EUCLID:56 ;
s1 in [.0 ,1.] by A15, XXREAL_1:1;
then A28: k . s1 = h . p1 by A15, A26, FUNCT_1:23
.= (proj1 . p1) + (proj2 . p1) by A22
.= (p1 `1 ) + (proj2 . p1) by PSCOMP_1:def 28
.= b + (p1 `2 ) by A3, PSCOMP_1:def 29 ;
A29: s2 in [.0 ,1.] by A15, XXREAL_1:1;
then k . s2 = h . p2 by A15, A26, FUNCT_1:23
.= (proj1 . p2) + (proj2 . p2) by A22
.= (p2 `1 ) + (proj2 . p2) by PSCOMP_1:def 28
.= b + (p2 `2 ) by A10, PSCOMP_1:def 29 ;
then A30: ( k . 0 >= k . s1 & k . s1 >= k . s2 ) by A3, A9, A27, A28, XREAL_1:9;
A31: 0 in [.0 ,1.] by XXREAL_1:1;
then A32: [.0 ,s2.] c= [.0 ,1.] by A29, XXREAL_2:def 12;
reconsider B = [.0 ,s2.] as Subset of I[01] by A29, A31, BORSUK_1:83, XXREAL_2:def 12;
A33: B is connected by A15, A29, A31, BORSUK_1:83, BORSUK_4:49;
A34: 0 in B by A15, XXREAL_1:1;
A35: s2 in B by A15, XXREAL_1:1;
A36: k . 0 is Real by XREAL_0:def 1;
A37: 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
A38: ( xc in B & k . xc = k . s1 ) by A23, A30, A33, A34, A35, A36, A37, TOPREAL5:11;
xc in [.0 ,1.] by BORSUK_1:83;
then reconsider rxc = xc as Real ;
A39: k is one-to-one
proof
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 A40: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 ) ; :: thesis: x1 = x2
then reconsider r1 = x1 as Point of I[01] ;
reconsider r2 = x2 as Point of I[01] by A40;
A41: k . x1 = h . (g1 . x1) by A40, FUNCT_1:22
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A22
.= ((g1 . r1) `1 ) + (proj2 . (g1 . r1)) by PSCOMP_1:def 28
.= ((g1 . r1) `1 ) + ((g1 . r1) `2 ) by PSCOMP_1:def 29 ;
A42: k . x2 = h . (g1 . x2) by A40, FUNCT_1:22
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A22
.= ((g1 . r2) `1 ) + (proj2 . (g1 . r2)) by PSCOMP_1:def 28
.= ((g1 . r2) `1 ) + ((g1 . r2) `2 ) by PSCOMP_1:def 29 ;
A43: g . r1 in Lower_Arc (rectangle a,b,c,d) by A18;
A44: g . r2 in Lower_Arc (rectangle a,b,c,d) by A18;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A43;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A44;
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 A12, A18, XBOOLE_0:def 3;
case A45: ( g . r1 in LSeg |[b,d]|,|[b,c]| & g . r2 in LSeg |[b,d]|,|[b,c]| ) ; :: thesis: x1 = x2
then A46: ( gr1 `1 = b & c <= gr1 `2 & gr1 `2 <= d ) by A1, Th9;
( gr2 `1 = b & c <= gr2 `2 & gr2 `2 <= d ) by A1, A45, Th9;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A40, A41, A42, A46, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A16, A17, FUNCT_1:def 8; :: thesis: verum
end;
case A47: ( g . r1 in LSeg |[b,d]|,|[b,c]| & g . r2 in LSeg |[b,c]|,|[a,c]| ) ; :: thesis: x1 = x2
then A48: ( gr1 `1 = b & c <= gr1 `2 & gr1 `2 <= d ) by A1, Th9;
A49: ( gr2 `2 = c & a <= gr2 `1 & gr2 `1 <= b ) by A1, A47, Th11;
then A50: b + (gr1 `2 ) = (gr2 `1 ) + c by A1, A40, A41, A42, A47, Th9;
A51: now
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by A49, XXREAL_0:1;
hence contradiction by A40, A41, A42, A48, A49, XREAL_1:10; :: thesis: verum
end;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A48, A49, A51, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A16, A17, FUNCT_1:def 8; :: thesis: verum
end;
case A52: ( g . r1 in LSeg |[b,c]|,|[a,c]| & g . r2 in LSeg |[b,d]|,|[b,c]| ) ; :: thesis: x1 = x2
then A53: ( gr2 `1 = b & c <= gr2 `2 & gr2 `2 <= d ) by A1, Th9;
A54: ( gr1 `2 = c & a <= gr1 `1 & gr1 `1 <= b ) by A1, A52, Th11;
A55: b + (gr2 `2 ) = (gr1 `1 ) + c by A1, A40, A41, A42, A52, A53, Th11;
A56: now
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by A54, XXREAL_0:1;
hence contradiction by A53, A55, XREAL_1:10; :: thesis: verum
end;
then |[(gr2 `1 ),(gr2 `2 )]| = g . r1 by A53, A54, A56, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A16, A17, FUNCT_1:def 8; :: thesis: verum
end;
case A57: ( g . r1 in LSeg |[b,c]|,|[a,c]| & g . r2 in LSeg |[b,c]|,|[a,c]| ) ; :: thesis: x1 = x2
then A58: ( gr1 `2 = c & a <= gr1 `1 & gr1 `1 <= b ) by A1, Th11;
( gr2 `2 = c & a <= gr2 `1 & gr2 `1 <= b ) by A1, A57, Th11;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A40, A41, A42, A58, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A16, A17, FUNCT_1:def 8; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
hence k is one-to-one by FUNCT_1:def 8; :: thesis: verum
end;
A59: dom k = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
then s1 in dom k by A15, XXREAL_1:1;
then rxc = s1 by A32, A38, A39, A59, FUNCT_1:def 8;
hence contradiction by A25, A38, XXREAL_1:1; :: thesis: verum
end;
hence s1 <= s2 ; :: thesis: verum
end;
hence 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 A13, A14, JORDAN5C:def 3; :: thesis: verum
end;
hence LE p1,p2, rectangle a,b,c,d by A11, A13, A14, JORDAN6:def 10; :: thesis: verum
end;
case A60: ( 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 A61: ( p2 `2 = c & a <= p2 `1 & p2 `1 <= b ) by A1, Th11;
A62: Lower_Arc (rectangle a,b,c,d) = (LSeg |[b,d]|,|[b,c]|) \/ (LSeg |[b,c]|,|[a,c]|) by A1, Th62;
then A63: p2 in Lower_Arc (rectangle a,b,c,d) by A60, XBOOLE_0:def 3;
A64: p1 in Lower_Arc (rectangle a,b,c,d) by A1, A62, XBOOLE_0:def 3;
LE p1,p2, Lower_Arc (rectangle a,b,c,d), E-max (rectangle a,b,c,d), W-min (rectangle a,b,c,d)
proof
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 A65: ( 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 ) ; :: thesis: s1 <= s2
A66: dom g = the carrier of I[01] by FUNCT_2:def 1;
A67: g is one-to-one by A65, TOPS_2:def 5;
A68: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) = Lower_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 A65, TOPS_2:def 5;
then A69: 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 ;
A70: 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 B24: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:39;
A71: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by B24, PRE_TOPC:62;
BB: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A70, JGRAPH_2:40;
( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by BB, PRE_TOPC:62;
then consider h being Function of (TOP-REAL 2),R^1 such that
A72: ( ( 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 ) & h is continuous ) by A71, JGRAPH_2:29;
reconsider k = h * g1 as Function of I[01] ,R^1 ;
A73: k is continuous Function of I[01] ,R^1 by A69, A72;
A74: E-max (rectangle a,b,c,d) = |[b,d]| by A1, Th56;
now
assume A75: s1 > s2 ; :: thesis: contradiction
A76: dom g = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
0 in [.0 ,1.] by XXREAL_1:1;
then A77: k . 0 = h . (E-max (rectangle a,b,c,d)) by A65, A76, FUNCT_1:23
.= (h1 . (E-max (rectangle a,b,c,d))) + (h2 . (E-max (rectangle a,b,c,d))) by A72
.= ((E-max (rectangle a,b,c,d)) `1 ) + (proj2 . (E-max (rectangle a,b,c,d))) by PSCOMP_1:def 28
.= ((E-max (rectangle a,b,c,d)) `1 ) + ((E-max (rectangle a,b,c,d)) `2 ) by PSCOMP_1:def 29
.= ((E-max (rectangle a,b,c,d)) `1 ) + d by A74, EUCLID:56
.= b + d by A74, EUCLID:56 ;
s1 in [.0 ,1.] by A65, XXREAL_1:1;
then A78: k . s1 = h . p1 by A65, A76, FUNCT_1:23
.= (proj1 . p1) + (proj2 . p1) by A72
.= (p1 `1 ) + (proj2 . p1) by PSCOMP_1:def 28
.= b + (p1 `2 ) by A3, PSCOMP_1:def 29 ;
A79: s2 in [.0 ,1.] by A65, XXREAL_1:1;
then k . s2 = h . p2 by A65, A76, FUNCT_1:23
.= (proj1 . p2) + (proj2 . p2) by A72
.= (p2 `1 ) + (proj2 . p2) by PSCOMP_1:def 28
.= (p2 `1 ) + c by A61, PSCOMP_1:def 29 ;
then A80: ( k . 0 >= k . s1 & k . s1 >= k . s2 ) by A3, A61, A77, A78, XREAL_1:9;
A81: 0 in [.0 ,1.] by XXREAL_1:1;
then A82: [.0 ,s2.] c= [.0 ,1.] by A79, XXREAL_2:def 12;
reconsider B = [.0 ,s2.] as Subset of I[01] by A79, A81, BORSUK_1:83, XXREAL_2:def 12;
A83: B is connected by A65, A79, A81, BORSUK_1:83, BORSUK_4:49;
A84: 0 in B by A65, XXREAL_1:1;
A85: s2 in B by A65, XXREAL_1:1;
A86: k . 0 is Real by XREAL_0:def 1;
A87: 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
A88: ( xc in B & k . xc = k . s1 ) by A73, A80, A83, A84, A85, A86, A87, TOPREAL5:11;
xc in [.0 ,1.] by BORSUK_1:83;
then reconsider rxc = xc as Real ;
A89: k is one-to-one
proof
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 A90: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 ) ; :: thesis: x1 = x2
then reconsider r1 = x1 as Point of I[01] ;
reconsider r2 = x2 as Point of I[01] by A90;
A91: k . x1 = h . (g1 . x1) by A90, FUNCT_1:22
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A72
.= ((g1 . r1) `1 ) + (proj2 . (g1 . r1)) by PSCOMP_1:def 28
.= ((g1 . r1) `1 ) + ((g1 . r1) `2 ) by PSCOMP_1:def 29 ;
A92: k . x2 = h . (g1 . x2) by A90, FUNCT_1:22
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A72
.= ((g1 . r2) `1 ) + (proj2 . (g1 . r2)) by PSCOMP_1:def 28
.= ((g1 . r2) `1 ) + ((g1 . r2) `2 ) by PSCOMP_1:def 29 ;
A93: g . r1 in Lower_Arc (rectangle a,b,c,d) by A68;
A94: g . r2 in Lower_Arc (rectangle a,b,c,d) by A68;
reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A93;
reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A94;
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 A62, A68, XBOOLE_0:def 3;
case A95: ( g . r1 in LSeg |[b,d]|,|[b,c]| & g . r2 in LSeg |[b,d]|,|[b,c]| ) ; :: thesis: x1 = x2
then A96: ( gr1 `1 = b & c <= gr1 `2 & gr1 `2 <= d ) by A1, Th9;
( gr2 `1 = b & c <= gr2 `2 & gr2 `2 <= d ) by A1, A95, Th9;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A90, A91, A92, A96, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A66, A67, FUNCT_1:def 8; :: thesis: verum
end;
case A97: ( g . r1 in LSeg |[b,d]|,|[b,c]| & g . r2 in LSeg |[b,c]|,|[a,c]| ) ; :: thesis: x1 = x2
then A98: ( gr1 `1 = b & c <= gr1 `2 & gr1 `2 <= d ) by A1, Th9;
A99: ( gr2 `2 = c & a <= gr2 `1 & gr2 `1 <= b ) by A1, A97, Th11;
then A100: b + (gr1 `2 ) = (gr2 `1 ) + c by A1, A90, A91, A92, A97, Th9;
A101: now
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by A99, XXREAL_0:1;
hence contradiction by A90, A91, A92, A98, A99, XREAL_1:10; :: thesis: verum
end;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A98, A99, A101, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A66, A67, FUNCT_1:def 8; :: thesis: verum
end;
case A102: ( g . r1 in LSeg |[b,c]|,|[a,c]| & g . r2 in LSeg |[b,d]|,|[b,c]| ) ; :: thesis: x1 = x2
then A103: ( gr2 `1 = b & c <= gr2 `2 & gr2 `2 <= d ) by A1, Th9;
A104: ( gr1 `2 = c & a <= gr1 `1 & gr1 `1 <= b ) by A1, A102, Th11;
A105: b + (gr2 `2 ) = (gr1 `1 ) + c by A1, A90, A91, A92, A102, A103, Th11;
A106: now
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by A104, XXREAL_0:1;
hence contradiction by A103, A105, XREAL_1:10; :: thesis: verum
end;
then |[(gr2 `1 ),(gr2 `2 )]| = g . r1 by A103, A104, A106, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A66, A67, FUNCT_1:def 8; :: thesis: verum
end;
case A107: ( g . r1 in LSeg |[b,c]|,|[a,c]| & g . r2 in LSeg |[b,c]|,|[a,c]| ) ; :: thesis: x1 = x2
then A108: ( gr1 `2 = c & a <= gr1 `1 & gr1 `1 <= b ) by A1, Th11;
( gr2 `2 = c & a <= gr2 `1 & gr2 `1 <= b ) by A1, A107, Th11;
then |[(gr1 `1 ),(gr1 `2 )]| = g . r2 by A90, A91, A92, A108, EUCLID:57;
then g . r1 = g . r2 by EUCLID:57;
hence x1 = x2 by A66, A67, FUNCT_1:def 8; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
hence k is one-to-one by FUNCT_1:def 8; :: thesis: verum
end;
A109: dom k = [.0 ,1.] by BORSUK_1:83, FUNCT_2:def 1;
then s1 in dom k by A65, XXREAL_1:1;
then rxc = s1 by A82, A88, A89, A109, FUNCT_1:def 8;
hence contradiction by A75, A88, XXREAL_1:1; :: thesis: verum
end;
hence s1 <= s2 ; :: thesis: verum
end;
hence 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 A63, A64, JORDAN5C:def 3; :: thesis: verum
end;
hence LE p1,p2, rectangle a,b,c,d by A60, A63, A64, JORDAN6:def 10; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle a,b,c,d ; :: thesis: verum
end;