let a, b, c, d be Real; :: thesis: for p1, p2 being Point of () st a < b & c < d & p1 in LSeg (|[b,c]|,|[a,c]|) & p1 <> W-min (rectangle (a,b,c,d)) holds
( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) )

let p1, p2 be Point of (); :: thesis: ( a < b & c < d & p1 in LSeg (|[b,c]|,|[a,c]|) & p1 <> W-min (rectangle (a,b,c,d)) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & 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,c]|,|[a,c]|) and
A4: p1 <> W-min (rectangle (a,b,c,d)) ; :: thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) )
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: ( 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) )
proof
assume A8: LE p1,p2, rectangle (a,b,c,d) ; :: thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) )
then A9: p2 in rectangle (a,b,c,d) by ;
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 ;
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: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & 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 ;
case p2 in LSeg (|[a,c]|,|[a,d]|) ; :: thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) )
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: verum
end;
case p2 in LSeg (|[a,d]|,|[b,d]|) ; :: thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) )
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: verum
end;
case p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) )
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: verum
end;
case p2 in LSeg (|[b,c]|,|[a,c]|) ; :: thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) )
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: verum
end;
end;
end;
hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ; :: thesis: verum
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: verum
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: LE p1,p2, rectangle (a,b,c,d)
now :: thesis: ( ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & 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 (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by ;
case A14: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 ) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A15: p2 `2 = c by ;
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 ;
A18: p1 in Lower_Arc (rectangle (a,b,c,d)) by ;
for g being Function of I[01],(() | (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],(() | (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
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: s1 <= s2
A27: dom g = the carrier of I[01] by FUNCT_2:def 1;
A28: g is one-to-one by ;
A29: the carrier of (() | (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],() by FUNCT_2:7;
g is continuous by ;
then A30: g1 is continuous by PRE_TOPC:26;
reconsider h1 = proj1 as Function of (),R^1 by TOPMETR:17;
reconsider h2 = proj2 as Function of (),R^1 by TOPMETR:17;
reconsider hh1 = h1 as Function of TopStruct(# the carrier of (), the topology of () #),R^1 ;
reconsider hh2 = h2 as Function of TopStruct(# the carrier of (), the topology of () #),R^1 ;
A31: TopStruct(# the carrier of (), the topology of () #) = TopStruct(# the carrier of (), the topology of () #) | ([#] TopStruct(# the carrier of (), the topology of () #)) by TSEP_1:3
.= TopStruct(# the carrier of (() | ([#] ())), the topology of (() | ([#] ())) #) by PRE_TOPC:36
.= () | ([#] ()) ;
then ( ( for p being Point of (() | ([#] ())) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29;
then A32: ( ( for p being Point of (() | ([#] ())) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32;
( ( for p being Point of (() | ([#] ())) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by ;
then ( ( for p being Point of (() | ([#] ())) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32;
then consider h being Function of (),R^1 such that
A33: for p being Point of ()
for r1, r2 being Real st h1 . p = r1 & h2 . p = r2 holds
h . p = r1 + r2 and
A34: h is continuous by ;
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: not s1 > s2
assume A36: s1 > s2 ; :: thesis: contradiction
A37: dom g = [.0,1.] by ;
0 in [.0,1.] by XXREAL_1:1;
then A38: k . 0 = h . (E-max (rectangle (a,b,c,d))) by
.= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A33
.= ((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
.= b + d by ;
s1 in [.0,1.] by ;
then A39: k . s1 = h . p1 by
.= (proj1 . p1) + (proj2 . p1) by A33
.= (p1 `1) + (proj2 . p1) by PSCOMP_1:def 5
.= (p1 `1) + c by ;
A40: s2 in [.0,1.] by ;
then A41: k . s2 = h . p2 by
.= (proj1 . p2) + (proj2 . p2) by A33
.= (p2 `1) + (proj2 . p2) by PSCOMP_1:def 5
.= (p2 `1) + c by ;
A42: k . 0 >= k . s1 by ;
A43: k . s1 >= k . s2 by ;
A44: 0 in [.0,1.] by XXREAL_1:1;
then A45: [.0,s2.] c= [.0,1.] by ;
reconsider B = [.0,s2.] as Subset of I[01] by ;
A46: B is connected by ;
A47: 0 in B by ;
A48: s2 in B by ;
consider xc being Point of I[01] such that
A49: xc in B and
A50: k . xc = k . s1 by ;
reconsider rxc = xc as Real ;
A51: 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
A52: x1 in dom k and
A53: x2 in dom k and
A54: k . x1 = k . x2 ; :: thesis: x1 = x2
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
.= (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
.= (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 () by A57;
reconsider gr2 = g . r2 as Point of () by A58;
now :: thesis: ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) & x1 = x2 ) )
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 ;
case A59: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A60: gr1 `1 = b by ;
gr2 `1 = b by A2, A59, Th1;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A61: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A62: gr1 `1 = b by ;
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: not b <> gr2 `1
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by ;
hence contradiction by A54, A55, A56, A62, A63, A64, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr1 `2 <> c
assume gr1 `2 <> c ; :: thesis: contradiction
then c < gr1 `2 by ;
hence contradiction by A65, A66, XREAL_1:8; :: thesis: verum
end;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A68: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A69: gr2 `1 = b by ;
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: not b <> gr1 `1
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by ;
hence contradiction by A70, A73, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr2 `2 <> c
assume gr2 `2 <> c ; :: thesis: contradiction
then c < gr2 `2 by ;
hence contradiction by A54, A55, A56, A69, A71, A72, XREAL_1:8; :: thesis: verum
end;
then |[(gr2 `1),(gr2 `2)]| = g . r1 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A75: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A76: gr1 `2 = c by ;
gr2 `2 = c by A1, A75, Th3;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
A77: dom k = [.0,1.] by ;
then s1 in dom k by ;
then rxc = s1 by A45, A49, A50, A51, A77;
hence contradiction by A36, A49, 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 ;
hence LE p1,p2, rectangle (a,b,c,d) by ; :: thesis: verum
end;
case A78: ( 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 A79: p2 `2 = c by ;
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 ;
A82: p1 in Lower_Arc (rectangle (a,b,c,d)) by ;
for g being Function of I[01],(() | (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],(() | (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
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: s1 <= s2
A91: dom g = the carrier of I[01] by FUNCT_2:def 1;
A92: g is one-to-one by ;
A93: the carrier of (() | (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],() by FUNCT_2:7;
g is continuous by ;
then A94: g1 is continuous by PRE_TOPC:26;
reconsider h1 = proj1 as Function of (),R^1 by TOPMETR:17;
reconsider h2 = proj2 as Function of (),R^1 by TOPMETR:17;
reconsider hh1 = h1 as Function of TopStruct(# the carrier of (), the topology of () #),R^1 ;
reconsider hh2 = h2 as Function of TopStruct(# the carrier of (), the topology of () #),R^1 ;
A95: TopStruct(# the carrier of (), the topology of () #) = TopStruct(# the carrier of (), the topology of () #) | ([#] TopStruct(# the carrier of (), the topology of () #)) by TSEP_1:3
.= TopStruct(# the carrier of (() | ([#] ())), the topology of (() | ([#] ())) #) by PRE_TOPC:36
.= () | ([#] ()) ;
then ( ( for p being Point of (() | ([#] ())) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29;
then A96: ( ( for p being Point of (() | ([#] ())) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32;
( ( for p being Point of (() | ([#] ())) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by ;
then ( ( for p being Point of (() | ([#] ())) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32;
then consider h being Function of (),R^1 such that
A97: for p being Point of ()
for r1, r2 being Real st h1 . p = r1 & h2 . p = r2 holds
h . p = r1 + r2 and
A98: h is continuous by ;
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: not s1 > s2
assume A100: s1 > s2 ; :: thesis: contradiction
A101: dom g = [.0,1.] by ;
0 in [.0,1.] by XXREAL_1:1;
then A102: k . 0 = h . (E-max (rectangle (a,b,c,d))) by
.= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A97
.= ((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
.= b + d by ;
s1 in [.0,1.] by ;
then A103: k . s1 = h . p1 by
.= (proj1 . p1) + (proj2 . p1) by A97
.= (p1 `1) + (proj2 . p1) by PSCOMP_1:def 5
.= (p1 `1) + c by ;
A104: s2 in [.0,1.] by ;
then A105: k . s2 = h . p2 by
.= (proj1 . p2) + (proj2 . p2) by A97
.= (p2 `1) + (proj2 . p2) by PSCOMP_1:def 5
.= (p2 `1) + c by ;
A106: k . 0 >= k . s1 by ;
A107: k . s1 >= k . s2 by ;
A108: 0 in [.0,1.] by XXREAL_1:1;
then A109: [.0,s2.] c= [.0,1.] by ;
reconsider B = [.0,s2.] as Subset of I[01] by ;
A110: B is connected by ;
A111: 0 in B by ;
A112: s2 in B by ;
consider xc being Point of I[01] such that
A113: xc in B and
A114: k . xc = k . s1 by ;
reconsider rxc = xc as Real ;
A115: 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
A116: x1 in dom k and
A117: x2 in dom k and
A118: k . x1 = k . x2 ; :: thesis: x1 = x2
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
.= (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
.= (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 () by A121;
reconsider gr2 = g . r2 as Point of () by A122;
now :: thesis: ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) & x1 = x2 ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) & x1 = x2 ) )
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 ;
case A123: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A124: gr1 `1 = b by ;
gr2 `1 = b by ;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A125: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A126: gr1 `1 = b by ;
A127: c <= gr1 `2 by ;
A128: gr2 `2 = c by ;
A129: gr2 `1 <= b by ;
A130: b + (gr1 `2) = (gr2 `1) + c by ;
A131: now :: thesis: not b <> gr2 `1
assume b <> gr2 `1 ; :: thesis: contradiction
then b > gr2 `1 by ;
hence contradiction by A118, A119, A120, A126, A127, A128, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr1 `2 <> c
assume gr1 `2 <> c ; :: thesis: contradiction
then c < gr1 `2 by ;
hence contradiction by A129, A130, XREAL_1:8; :: thesis: verum
end;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A132: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; :: thesis: x1 = x2
then A133: gr2 `1 = b by ;
A134: c <= gr2 `2 by ;
A135: gr1 `2 = c by ;
A136: gr1 `1 <= b by ;
A137: b + (gr2 `2) = (gr1 `1) + c by ;
A138: now :: thesis: not b <> gr1 `1
assume b <> gr1 `1 ; :: thesis: contradiction
then b > gr1 `1 by ;
hence contradiction by A134, A137, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr2 `2 <> cend;
then |[(gr2 `1),(gr2 `2)]| = g . r1 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
case A139: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; :: thesis: x1 = x2
then A140: gr1 `2 = c by ;
gr2 `2 = c by ;
then |[(gr1 `1),(gr1 `2)]| = g . r2 by ;
then g . r1 = g . r2 by EUCLID:53;
hence x1 = x2 by ; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
A141: dom k = [.0,1.] by ;
then s1 in dom k by ;
then rxc = s1 by ;
hence contradiction by A100, A113, 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 ;
hence LE p1,p2, rectangle (a,b,c,d) by ; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle (a,b,c,d) ; :: thesis: verum
end;