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 & b,a,c,d are_in_this_order_on P holds
a = b
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 & b,a,c,d are_in_this_order_on P implies a = b )
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: b,a,c,d are_in_this_order_on P ; :: thesis: a = b
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: a = b
then A7: LE b,d,P by JORDAN6:58;
thus a = b :: thesis: verum
proof
per cases ( ( LE b,a,P & LE a,c,P & LE c,d,P ) or ( LE a,c,P & LE c,d,P & LE d,b,P ) or ( LE c,d,P & LE d,b,P & LE b,a,P ) or ( LE d,b,P & LE b,a,P & LE a,c,P ) ) by A5;
suppose ( LE b,a,P & LE a,c,P & LE c,d,P ) ; :: thesis: a = b
hence a = b by A6, JORDAN6:57; :: thesis: verum
end;
suppose ( LE a,c,P & LE c,d,P & LE d,b,P ) ; :: thesis: a = b
hence a = b by A3, A7, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,d,P & LE d,b,P & LE b,a,P ) ; :: thesis: a = b
hence a = b by A6, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,b,P & LE b,a,P & LE a,c,P ) ; :: thesis: a = b
hence a = b by A6, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A8: ( LE b,c,P & LE c,d,P & LE d,a,P ) ; :: thesis: a = b
then A9: LE b,d,P by JORDAN6:58;
thus a = b :: thesis: verum
proof
per cases ( ( LE b,a,P & LE a,c,P & LE c,d,P ) or ( LE a,c,P & LE c,d,P & LE d,b,P ) or ( LE c,d,P & LE d,b,P & LE b,a,P ) or ( LE d,b,P & LE b,a,P & LE a,c,P ) ) by A5;
suppose A10: ( LE b,a,P & LE a,c,P & LE c,d,P ) ; :: thesis: a = b
LE c,a,P by A8, JORDAN6:58;
hence a = b by A1, A10, JORDAN6:57; :: thesis: verum
end;
suppose ( LE a,c,P & LE c,d,P & LE d,b,P ) ; :: thesis: a = b
hence a = b by A3, A9, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,d,P & LE d,b,P & LE b,a,P ) ; :: thesis: a = b
hence a = b by A3, A9, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,b,P & LE b,a,P & LE a,c,P ) ; :: thesis: a = b
hence a = b by A3, A9, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A11: ( LE c,d,P & LE d,a,P & LE a,b,P ) ; :: thesis: a = b
then A12: LE c,a,P by JORDAN6:58;
thus a = b :: thesis: verum
proof
per cases ( ( LE b,a,P & LE a,c,P & LE c,d,P ) or ( LE a,c,P & LE c,d,P & LE d,b,P ) or ( LE c,d,P & LE d,b,P & LE b,a,P ) or ( LE d,b,P & LE b,a,P & LE a,c,P ) ) by A5;
suppose ( LE b,a,P & LE a,c,P & LE c,d,P ) ; :: thesis: a = b
hence a = b by A11, JORDAN6:57; :: thesis: verum
end;
suppose ( LE a,c,P & LE c,d,P & LE d,b,P ) ; :: thesis: a = b
hence a = b by A1, A12, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,d,P & LE d,b,P & LE b,a,P ) ; :: thesis: a = b
hence a = b by A11, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,b,P & LE b,a,P & LE a,c,P ) ; :: thesis: a = b
hence a = b by A11, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A13: ( LE d,a,P & LE a,b,P & LE b,c,P ) ; :: thesis: a = b
thus a = b :: thesis: verum
proof
per cases ( ( LE b,a,P & LE a,c,P & LE c,d,P ) or ( LE a,c,P & LE c,d,P & LE d,b,P ) or ( LE c,d,P & LE d,b,P & LE b,a,P ) or ( LE d,b,P & LE b,a,P & LE a,c,P ) ) by A5;
suppose ( LE b,a,P & LE a,c,P & LE c,d,P ) ; :: thesis: a = b
hence a = b by A13, JORDAN6:57; :: thesis: verum
end;
suppose ( LE a,c,P & LE c,d,P & LE d,b,P ) ; :: thesis: a = b
then LE a,d,P by JORDAN6:58;
hence a = b by A2, A13, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,d,P & LE d,b,P & LE b,a,P ) ; :: thesis: a = b
hence a = b by A13, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,b,P & LE b,a,P & LE a,c,P ) ; :: thesis: a = b
hence a = b by A13, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
end;