let P be Subset of (TOP-REAL 2); :: thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q1,P holds

q1 = q2

let q1, q2 be Point of (TOP-REAL 2); :: thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q1,P implies q1 = q2 )

assume that

A1: P is being_simple_closed_curve and

A2: LE q1,q2,P and

A3: LE q2,q1,P ; :: thesis: q1 = q2

A4: Lower_Arc P is_an_arc_of E-max P, W-min P by A1, Def9;

A5: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A1, Def9;

A6: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, Th50;

q1 = q2

let q1, q2 be Point of (TOP-REAL 2); :: thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q1,P implies q1 = q2 )

assume that

A1: P is being_simple_closed_curve and

A2: LE q1,q2,P and

A3: LE q2,q1,P ; :: thesis: q1 = q2

A4: Lower_Arc P is_an_arc_of E-max P, W-min P by A1, Def9;

A5: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A1, Def9;

A6: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, Th50;

now :: thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P & q1 = q2 ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P & q1 = q2 ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P & q1 = q2 ) )end;

hence
q1 = q2
; :: thesis: verumper cases
( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
by A2;

end;

case A7:
( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P )
; :: thesis: q1 = q2

end;

now :: thesis: ( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P & q1 = q2 ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P & q1 = q2 ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P & q1 = q2 ) )end;

hence
q1 = q2
; :: thesis: verumper cases
( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P ) )
by A3;

end;

case A8:
( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P )
; :: thesis: q1 = q2

then
q1 in (Upper_Arc P) /\ (Lower_Arc P)
by A7, XBOOLE_0:def 4;

then A9: q1 = E-max P by A5, A8, TARSKI:def 2;

q2 in (Upper_Arc P) /\ (Lower_Arc P) by A7, A8, XBOOLE_0:def 4;

hence q1 = q2 by A5, A7, A9, TARSKI:def 2; :: thesis: verum

end;then A9: q1 = E-max P by A5, A8, TARSKI:def 2;

q2 in (Upper_Arc P) /\ (Lower_Arc P) by A7, A8, XBOOLE_0:def 4;

hence q1 = q2 by A5, A7, A9, TARSKI:def 2; :: thesis: verum

case A12:
( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P )
; :: thesis: q1 = q2

end;

now :: thesis: ( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P & q1 = q2 ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P & q1 = q2 ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P & q1 = q2 ) )end;

hence
q1 = q2
; :: thesis: verumper cases
( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P ) )
by A3;

end;

case A13:
( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P )
; :: thesis: q1 = q2

then
q1 in (Upper_Arc P) /\ (Lower_Arc P)
by A12, XBOOLE_0:def 4;

then q1 = E-max P by A5, A13, TARSKI:def 2;

hence q1 = q2 by A6, A12, Th55; :: thesis: verum

end;then q1 = E-max P by A5, A13, TARSKI:def 2;

hence q1 = q2 by A6, A12, Th55; :: thesis: verum

case A15:
( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P )
; :: thesis: q1 = q2

end;

now :: thesis: ( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P & q1 = q2 ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P & q1 = q2 ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P & q1 = q2 ) )end;

hence
q1 = q2
; :: thesis: verumper cases
( ( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P ) or ( q2 in Upper_Arc P & q1 in Upper_Arc P & LE q2,q1, Upper_Arc P, W-min P, E-max P ) or ( q2 in Lower_Arc P & q1 in Lower_Arc P & not q1 = W-min P & LE q2,q1, Lower_Arc P, E-max P, W-min P ) )
by A3;

end;

case
( q2 in Upper_Arc P & q1 in Lower_Arc P & not q1 = W-min P )
; :: thesis: q1 = q2

then
q2 in (Upper_Arc P) /\ (Lower_Arc P)
by A15, XBOOLE_0:def 4;

then q2 = E-max P by A5, A15, TARSKI:def 2;

hence q1 = q2 by A4, A15, Th54; :: thesis: verum

end;then q2 = E-max P by A5, A15, TARSKI:def 2;

hence q1 = q2 by A4, A15, Th54; :: thesis: verum