let a, b, c, d be Real; :: thesis: for p1, p2 being Point of () 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 (); :: 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, Th50;
Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;
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, Th1;
A7: c <= p1 `2 by A2, A3, Th1;
A8: p1 `2 <= d by A2, A3, Th1;
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 ;
A11: 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 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 :: thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & ( ( 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)) ) ) ) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & ( ( 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)) ) ) ) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & ( ( 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)) ) ) ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & ( ( 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)) ) ) ) )
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 (|[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, Th55; :: 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 :: thesis: ( ( p2 = W-min (rectangle (a,b,c,d)) & ( ( 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)) ) ) ) or ( p2 <> W-min (rectangle (a,b,c,d)) & ( ( 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)) ) ) ) )
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 ;
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 ; :: 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, Th46;
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 :: thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 & LE p1,p2, rectangle (a,b,c,d) ) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & LE p1,p2, rectangle (a,b,c,d) ) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & 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 (|[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, Th55; :: 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 ;
A18: a <= p2 `1 by A1, A16, Th3;
A19: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;
then A20: p2 in Upper_Arc (rectangle (a,b,c,d)) by ;
A21: p1 in Upper_Arc (rectangle (a,b,c,d)) by ;
for g being Function of I[01],(() | (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],(() | (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 ;
A32: the carrier of (() | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_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 A33: 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 ;
A34: 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 A35: 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
A36: for p being Point of ()
for r1, r2 being Real st h1 . p = r1 & h2 . p = r2 holds
h . p = r1 + r2 and
A37: h is continuous by ;
reconsider k = h * g1 as Function of I[01],R^1 ;
A38: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46;
now :: thesis: not s1 > s2
assume A39: s1 > s2 ; :: thesis: contradiction
A40: dom g = [.0,1.] by ;
0 in [.0,1.] by XXREAL_1:1;
then A41: k . 0 = h . (W-min (rectangle (a,b,c,d))) by
.= (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 5
.= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def 6
.= ((W-min (rectangle (a,b,c,d))) `1) + c by
.= a + c by ;
s1 in [.0,1.] by ;
then A42: k . s1 = h . p1 by
.= (proj1 . p1) + (proj2 . p1) by A36
.= (p1 `1) + (proj2 . p1) by PSCOMP_1:def 5
.= a + (p1 `2) by ;
A43: s2 in [.0,1.] by ;
then A44: k . s2 = h . p2 by
.= (proj1 . p2) + (proj2 . p2) by A36
.= (p2 `1) + (proj2 . p2) by PSCOMP_1:def 5
.= (p2 `1) + d by ;
A45: k . 0 <= k . s1 by ;
A46: k . s1 <= k . s2 by ;
A47: 0 in [.0,1.] by XXREAL_1:1;
then A48: [.0,s2.] c= [.0,1.] by ;
reconsider B = [.0,s2.] as Subset of I[01] by ;
A49: B is connected by ;
A50: 0 in B by ;
A51: s2 in B by ;
consider xc being Point of I[01] such that
A52: xc in B and
A53: k . xc = k . s1 by ;
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
.= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A36
.= ((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
.= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A36
.= ((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 Upper_Arc (rectangle (a,b,c,d)) by A32;
A61: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A32;
reconsider gr1 = g . r1 as Point of () by A60;
reconsider gr2 = g . r2 as Point of () by A61;
now :: thesis: ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) & x1 = x2 ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) & x1 = x2 ) )
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 ;
case A62: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; :: thesis: x1 = x2
then A63: gr1 `1 = a by ;
gr2 `1 = a by A2, A62, 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 A64: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; :: thesis: x1 = x2
then A65: gr1 `1 = a by ;
A66: gr1 `2 <= d by A2, A64, Th1;
A67: gr2 `2 = d by A1, A64, Th3;
A68: a <= gr2 `1 by A1, A64, Th3;
A69: a + (gr1 `2) = (gr2 `1) + d by A1, A57, A58, A59, A64, A65, Th3;
A70: now :: thesis: not a <> gr2 `1
assume a <> gr2 `1 ; :: thesis: contradiction
then a < gr2 `1 by ;
hence contradiction by A66, A69, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr1 `2 <> d
assume gr1 `2 <> d ; :: thesis: contradiction
then d > gr1 `2 by ;
hence contradiction by A57, A58, A59, A65, A67, A68, 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 A71: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; :: thesis: x1 = x2
then A72: gr2 `1 = a by ;
A73: gr2 `2 <= d by A2, A71, Th1;
A74: gr1 `2 = d by A1, A71, Th3;
A75: a <= gr1 `1 by A1, A71, Th3;
A76: a + (gr2 `2) = (gr1 `1) + d by A1, A57, A58, A59, A71, A72, Th3;
A77: now :: thesis: not a <> gr1 `1
assume a <> gr1 `1 ; :: thesis: contradiction
then a < gr1 `1 by ;
hence contradiction by A73, A76, XREAL_1:8; :: thesis: verum
end;
now :: thesis: not gr2 `2 <> d
assume gr2 `2 <> d ; :: thesis: contradiction
then d > gr2 `2 by ;
hence contradiction by A57, A58, A59, A72, A74, A75, 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 A78: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; :: thesis: x1 = x2
then A79: gr1 `2 = d by ;
gr2 `2 = d by A1, A78, 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;
A80: dom k = [.0,1.] by ;
then s1 in dom k by ;
then rxc = s1 by A48, A52, A53, A54, A80;
hence contradiction by A39, A52, 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 ;
hence LE p1,p2, rectangle (a,b,c,d) by ; :: thesis: verum
end;
case A81: p2 in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: LE p1,p2, rectangle (a,b,c,d)
then A82: p2 `1 = b by TOPREAL3:11;
Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
then A83: LSeg (|[b,d]|,|[b,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;
p2 <> W-min (rectangle (a,b,c,d)) by ;
hence LE p1,p2, rectangle (a,b,c,d) by ; :: thesis: verum
end;
case A84: ( 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, Th52;
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 ; :: thesis: verum
end;
end;
end;
hence LE p1,p2, rectangle (a,b,c,d) ; :: thesis: verum
end;