let a, b, c, d be Real; :: thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) holds

( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 (TOP-REAL 2); :: thesis: ( a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) ; :: thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;

A6: p1 `2 = d by A1, A3, Th3;

A7: a <= p1 `1 by A1, A3, Th3;

thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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) )

thus ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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

( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 (TOP-REAL 2); :: thesis: ( a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) ; :: thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7;

A6: p1 `2 = d by A1, A3, Th3;

A7: a <= p1 `1 by A1, A3, Th3;

thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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

A13:
W-min (rectangle (a,b,c,d)) = |[a,c]|
by A1, A2, Th46;
assume A8:
LE p1,p2, rectangle (a,b,c,d)
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 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;

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

hence
( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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: verumper 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;

end;

case
p2 in LSeg (|[a,c]|,|[a,d]|)
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A1, A2, A3, Th59;

hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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, A8, Th50, JORDAN6:57; :: thesis: verum

end;hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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, A8, Th50, JORDAN6:57; :: thesis: verum

case
p2 in LSeg (|[a,d]|,|[b,d]|)
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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, A8, Th56; :: thesis: verum

end;case
p2 in LSeg (|[b,d]|,|[b,c]|)
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A12:
p2 in LSeg (|[b,c]|,|[a,c]|)
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) )

end;

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

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

end;

case
p2 = W-min (rectangle (a,b,c,d))
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A4, A9, JORDAN7:3;

hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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, A8, Th50, JORDAN6:57; :: thesis: verum

end;hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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, A8, Th50, JORDAN6:57; :: thesis: verum

case
p2 <> W-min (rectangle (a,b,c,d))
; :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A12; :: thesis: verum

end;thus ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A14:
( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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)

end;now :: thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 & 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) ) )end;

hence
LE p1,p2, rectangle (a,b,c,d)
; :: thesis: verumper cases
( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) 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 A14;

end;

case A15:
( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 )
; :: thesis: LE p1,p2, rectangle (a,b,c,d)

then A16:
p2 `2 = d
by A1, Th3;

A17: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;

then A18: p2 in Upper_Arc (rectangle (a,b,c,d)) by A15, XBOOLE_0:def 3;

A19: p1 in Upper_Arc (rectangle (a,b,c,d)) by A3, A17, XBOOLE_0:def 3;

for g being Function of I[01],((TOP-REAL 2) | (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

hence LE p1,p2, rectangle (a,b,c,d) by A18, A19, JORDAN6:def 10; :: thesis: verum

end;A17: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51;

then A18: p2 in Upper_Arc (rectangle (a,b,c,d)) by A15, XBOOLE_0:def 3;

A19: p1 in Upper_Arc (rectangle (a,b,c,d)) by A3, A17, XBOOLE_0:def 3;

for g being Function of I[01],((TOP-REAL 2) | (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

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 A18, A19, JORDAN5C:def 3;
let g be Function of I[01],((TOP-REAL 2) | (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

A20: g is being_homeomorphism and

A21: g . 0 = W-min (rectangle (a,b,c,d)) and

g . 1 = E-max (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) | (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],(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 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: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46;

end;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

A20: g is being_homeomorphism and

A21: g . 0 = W-min (rectangle (a,b,c,d)) and

g . 1 = E-max (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) | (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],(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 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: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46;

now :: thesis: not s1 > s2

hence
s1 <= s2
; :: thesis: verumassume 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 . (W-min (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13

.= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A34

.= ((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 A36, EUCLID:52

.= a + c 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

.= (p1 `1) + d by A6, 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

.= (p2 `1) + d by A16, PSCOMP_1:def 6 ;

A43: k . 0 <= k . s1 by A2, A7, A39, A40, XREAL_1:7;

A44: k . s1 <= k . s2 by A15, 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;

consider xc being Point of I[01] such that

A50: xc in B and

A51: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, TOPREAL5:5;

reconsider rxc = xc as Real ;

A52: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds

x1 = x2

then s1 in dom k by A23, A24, XXREAL_1:1;

then rxc = s1 by A46, A50, A51, A52, A78;

hence contradiction by A37, A50, XXREAL_1:1; :: thesis: verum

end;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 . (W-min (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13

.= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A34

.= ((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 A36, EUCLID:52

.= a + c 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

.= (p1 `1) + d by A6, 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

.= (p2 `1) + d by A16, PSCOMP_1:def 6 ;

A43: k . 0 <= k . s1 by A2, A7, A39, A40, XREAL_1:7;

A44: k . s1 <= k . s2 by A15, 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;

consider xc being Point of I[01] such that

A50: xc in B and

A51: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, TOPREAL5:5;

reconsider rxc = xc as Real ;

A52: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds

x1 = x2

proof

A78:
dom k = [.0,1.]
by BORSUK_1:40, FUNCT_2:def 1;
let x1, x2 be set ; :: thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 )

assume that

A53: x1 in dom k and

A54: x2 in dom k and

A55: k . x1 = k . x2 ; :: thesis: x1 = x2

reconsider r1 = x1 as Point of I[01] by A53;

reconsider r2 = x2 as Point of I[01] by A54;

A56: k . x1 = h . (g1 . x1) by A53, 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 ;

A57: k . x2 = h . (g1 . x2) by A54, 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 ;

A58: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A30;

A59: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A30;

reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A58;

reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A59;

end;assume that

A53: x1 in dom k and

A54: x2 in dom k and

A55: k . x1 = k . x2 ; :: thesis: x1 = x2

reconsider r1 = x1 as Point of I[01] by A53;

reconsider r2 = x2 as Point of I[01] by A54;

A56: k . x1 = h . (g1 . x1) by A53, 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 ;

A57: k . x2 = h . (g1 . x2) by A54, 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 ;

A58: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A30;

A59: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A30;

reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A58;

reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A59;

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 ) )end;

hence
x1 = x2
; :: thesis: verumper 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 A17, A30, XBOOLE_0:def 3;

end;

case A60:
( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) )
; :: thesis: x1 = x2

then A61:
gr1 `1 = a
by A2, Th1;

gr2 `1 = a by A2, A60, Th1;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A61, EUCLID:53;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

end;gr2 `1 = a by A2, A60, Th1;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A61, EUCLID:53;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

case A62:
( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) )
; :: thesis: x1 = x2

then A63:
gr1 `1 = a
by A2, Th1;

A64: gr1 `2 <= d by A2, A62, Th1;

A65: gr2 `2 = d by A1, A62, Th3;

A66: a <= gr2 `1 by A1, A62, Th3;

A67: a + (gr1 `2) = (gr2 `1) + d by A1, A55, A56, A57, A62, A63, Th3;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

end;A64: gr1 `2 <= d by A2, A62, Th1;

A65: gr2 `2 = d by A1, A62, Th3;

A66: a <= gr2 `1 by A1, A62, Th3;

A67: a + (gr1 `2) = (gr2 `1) + d by A1, A55, A56, A57, A62, A63, Th3;

A68: now :: thesis: not a <> gr2 `1

assume
a <> gr2 `1
; :: thesis: contradiction

then a < gr2 `1 by A66, XXREAL_0:1;

hence contradiction by A64, A67, XREAL_1:8; :: thesis: verum

end;then a < gr2 `1 by A66, XXREAL_0:1;

hence contradiction by A64, A67, XREAL_1:8; :: thesis: verum

now :: thesis: not gr1 `2 <> d

then
|[(gr1 `1),(gr1 `2)]| = g . r2
by A63, A65, A68, EUCLID:53;assume
gr1 `2 <> d
; :: thesis: contradiction

then d > gr1 `2 by A64, XXREAL_0:1;

hence contradiction by A55, A56, A57, A63, A65, A66, XREAL_1:8; :: thesis: verum

end;then d > gr1 `2 by A64, XXREAL_0:1;

hence contradiction by A55, A56, A57, A63, A65, A66, XREAL_1:8; :: thesis: verum

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

case A69:
( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) )
; :: thesis: x1 = x2

then A70:
gr2 `1 = a
by A2, Th1;

A71: gr2 `2 <= d by A2, A69, Th1;

A72: gr1 `2 = d by A1, A69, Th3;

A73: a <= gr1 `1 by A1, A69, Th3;

A74: a + (gr2 `2) = (gr1 `1) + d by A1, A55, A56, A57, A69, A70, Th3;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

end;A71: gr2 `2 <= d by A2, A69, Th1;

A72: gr1 `2 = d by A1, A69, Th3;

A73: a <= gr1 `1 by A1, A69, Th3;

A74: a + (gr2 `2) = (gr1 `1) + d by A1, A55, A56, A57, A69, A70, Th3;

A75: now :: thesis: not a <> gr1 `1

assume
a <> gr1 `1
; :: thesis: contradiction

then a < gr1 `1 by A73, XXREAL_0:1;

hence contradiction by A71, A74, XREAL_1:8; :: thesis: verum

end;then a < gr1 `1 by A73, XXREAL_0:1;

hence contradiction by A71, A74, XREAL_1:8; :: thesis: verum

now :: thesis: not gr2 `2 <> d

then
|[(gr2 `1),(gr2 `2)]| = g . r1
by A70, A72, A75, EUCLID:53;assume
gr2 `2 <> d
; :: thesis: contradiction

then d > gr2 `2 by A71, XXREAL_0:1;

hence contradiction by A55, A56, A57, A70, A72, A73, XREAL_1:8; :: thesis: verum

end;then d > gr2 `2 by A71, XXREAL_0:1;

hence contradiction by A55, A56, A57, A70, A72, A73, XREAL_1:8; :: thesis: verum

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

case A76:
( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) )
; :: thesis: x1 = x2

then A77:
gr1 `2 = d
by A1, Th3;

gr2 `2 = d by A1, A76, Th3;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A77, EUCLID:53;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

end;gr2 `2 = d by A1, A76, Th3;

then |[(gr1 `1),(gr1 `2)]| = g . r2 by A55, A56, A57, A77, EUCLID:53;

then g . r1 = g . r2 by EUCLID:53;

hence x1 = x2 by A28, A29, FUNCT_1:def 4; :: thesis: verum

then s1 in dom k by A23, A24, XXREAL_1:1;

then rxc = s1 by A46, A50, A51, A52, A78;

hence contradiction by A37, A50, XXREAL_1:1; :: thesis: verum

hence LE p1,p2, rectangle (a,b,c,d) by A18, A19, JORDAN6:def 10; :: thesis: verum

case A79:
p2 in LSeg (|[b,d]|,|[b,c]|)
; :: thesis: LE p1,p2, rectangle (a,b,c,d)

then A80:
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 A81: 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 A1, A13, A80, EUCLID:52;

hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A79, A81, JORDAN6:def 10; :: thesis: verum

end;Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;

then A81: 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 A1, A13, A80, EUCLID:52;

hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A79, A81, JORDAN6:def 10; :: thesis: verum

case A82:
( 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 A3, A5, A82, JORDAN6:def 10; :: thesis: verum

end;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 A3, A5, A82, JORDAN6:def 10; :: thesis: verum