let P be Simple_closed_curve; :: thesis: for a, b, c, d being Point of (TOP-REAL 2) st a <> c & a <> d & b <> d & a,b,c,d are_in_this_order_on P & a,c,b,d are_in_this_order_on P holds
b = c
let a, b, c, d be Point of (TOP-REAL 2); :: thesis: ( a <> c & a <> d & b <> d & a,b,c,d are_in_this_order_on P & a,c,b,d are_in_this_order_on P implies b = c )
assume that
A1: a <> c and
A2: a <> d and
A3: b <> d and
A4: a,b,c,d are_in_this_order_on P and
A5: a,c,b,d are_in_this_order_on P ; :: thesis: b = c

per cases ( ( LE a,b,P & LE b,c,P & LE c,d,P ) or ( LE b,c,P & LE c,d,P & LE d,a,P ) or ( LE c,d,P & LE d,a,P & LE a,b,P ) or ( LE d,a,P & LE a,b,P & LE b,c,P ) ) by A4;

suppose A6: ( LE a,b,P & LE b,c,P & LE c,d,P ) ; :: thesis: b = c

then A7: LE b,d,P by JORDAN6:58;
thus b = c :: thesis: verum

proof

per cases ( ( LE a,c,P & LE c,b,P & LE b,d,P ) or ( LE c,b,P & LE b,d,P & LE d,a,P ) or ( LE b,d,P & LE d,a,P & LE a,c,P ) or ( LE d,a,P & LE a,c,P & LE c,b,P ) ) by A5;

suppose ( LE a,c,P & LE c,b,P & LE b,d,P ) ; :: thesis: b = c

hence b = c by A6, JORDAN6:57; :: thesis: verum

end;

suppose ( LE c,b,P & LE b,d,P & LE d,a,P ) ; :: thesis: b = c

hence b = c by A6, JORDAN6:57; :: thesis: verum

end;

suppose A8: ( LE b,d,P & LE d,a,P & LE a,c,P ) ; :: thesis: b = c

LE a,d,P by A6, A7, JORDAN6:58;
hence b = c by A2, A8, JORDAN6:57; :: thesis: verum

end;

suppose ( LE d,a,P & LE a,c,P & LE c,b,P ) ; :: thesis: b = c

hence b = c by A6, JORDAN6:57; :: thesis: verum

end;

end;

end;

end;

suppose A9: ( LE b,c,P & LE c,d,P & LE d,a,P ) ; :: thesis: b = c

then A10: LE c,a,P by JORDAN6:58;
thus b = c :: thesis: verum

proof

per cases ( ( LE a,c,P & LE c,b,P & LE b,d,P ) or ( LE c,b,P & LE b,d,P & LE d,a,P ) or ( LE b,d,P & LE d,a,P & LE a,c,P ) or ( LE d,a,P & LE a,c,P & LE c,b,P ) ) by A5;

suppose ( LE a,c,P & LE c,b,P & LE b,d,P ) ; :: thesis: b = c

hence b = c by A9, JORDAN6:57; :: thesis: verum

end;

suppose ( LE c,b,P & LE b,d,P & LE d,a,P ) ; :: thesis: b = c

hence b = c by A9, JORDAN6:57; :: thesis: verum

end;

suppose ( LE b,d,P & LE d,a,P & LE a,c,P ) ; :: thesis: b = c

hence b = c by A1, A10, JORDAN6:57; :: thesis: verum

end;

suppose ( LE d,a,P & LE a,c,P & LE c,b,P ) ; :: thesis: b = c

hence b = c by A9, JORDAN6:57; :: thesis: verum

end;

end;

end;

end;

suppose A11: ( LE c,d,P & LE d,a,P & LE a,b,P ) ; :: thesis: b = c

then A12: LE c,a,P by JORDAN6:58;
thus b = c :: thesis: verum

proof

per cases ( ( LE a,c,P & LE c,b,P & LE b,d,P ) or ( LE c,b,P & LE b,d,P & LE d,a,P ) or ( LE b,d,P & LE d,a,P & LE a,c,P ) or ( LE d,a,P & LE a,c,P & LE c,b,P ) ) by A5;

suppose ( LE a,c,P & LE c,b,P & LE b,d,P ) ; :: thesis: b = c

hence b = c by A1, A12, JORDAN6:57; :: thesis: verum

end;

suppose A13: ( LE c,b,P & LE b,d,P & LE d,a,P ) ; :: thesis: b = c

LE d,b,P by A11, JORDAN6:58;
hence b = c by A3, A13, JORDAN6:57; :: thesis: verum

end;

suppose ( LE b,d,P & LE d,a,P & LE a,c,P ) ; :: thesis: b = c

hence b = c by A1, A12, JORDAN6:57; :: thesis: verum

end;

suppose ( LE d,a,P & LE a,c,P & LE c,b,P ) ; :: thesis: b = c

hence b = c by A1, A12, JORDAN6:57; :: thesis: verum

end;

end;

end;

end;

suppose A14: ( LE d,a,P & LE a,b,P & LE b,c,P ) ; :: thesis: b = c

thus b = c :: thesis: verum

proof

per cases ( ( LE a,c,P & LE c,b,P & LE b,d,P ) or ( LE c,b,P & LE b,d,P & LE d,a,P ) or ( LE b,d,P & LE d,a,P & LE a,c,P ) or ( LE d,a,P & LE a,c,P & LE c,b,P ) ) by A5;

suppose ( LE a,c,P & LE c,b,P & LE b,d,P ) ; :: thesis: b = c

hence b = c by A14, JORDAN6:57; :: thesis: verum

end;

suppose ( LE c,b,P & LE b,d,P & LE d,a,P ) ; :: thesis: b = c

hence b = c by A14, JORDAN6:57; :: thesis: verum

end;

suppose A15: ( LE b,d,P & LE d,a,P & LE a,c,P ) ; :: thesis: b = c

LE d,b,P by A14, JORDAN6:58;
hence b = c by A3, A15, JORDAN6:57; :: thesis: verum

end;

suppose ( LE d,a,P & LE a,c,P & LE c,b,P ) ; :: thesis: b = c

hence b = c by A14, JORDAN6:57; :: thesis: verum

end;

end;

end;

end;

end;