let P be Simple_closed_curve; :: thesis: for a, b, c, d being Point of (TOP-REAL 2) st a <> b & a <> c & b <> d & a,b,c,d are_in_this_order_on P & a,b,d,c are_in_this_order_on P holds
c = d
let a, b, c, d be Point of (TOP-REAL 2); :: thesis: ( a <> b & a <> c & b <> d & a,b,c,d are_in_this_order_on P & a,b,d,c are_in_this_order_on P implies c = d )
assume that
A1: a <> b and
A2: a <> c and
A3: b <> d and
A4: a,b,c,d are_in_this_order_on P and
A5: a,b,d,c are_in_this_order_on P ; :: thesis: c = d
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: c = d
thus c = d :: thesis: verum
proof
per cases ( ( LE a,b,P & LE b,d,P & LE d,c,P ) or ( LE b,d,P & LE d,c,P & LE c,a,P ) or ( LE d,c,P & LE c,a,P & LE a,b,P ) or ( LE c,a,P & LE a,b,P & LE b,d,P ) ) by A5;
suppose ( LE a,b,P & LE b,d,P & LE d,c,P ) ; :: thesis: c = d
hence c = d by A6, JORDAN6:57; :: thesis: verum
end;
suppose ( LE b,d,P & LE d,c,P & LE c,a,P ) ; :: thesis: c = d
hence c = d by A6, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,c,P & LE c,a,P & LE a,b,P ) ; :: thesis: c = d
hence c = d by A6, JORDAN6:57; :: thesis: verum
end;
suppose A7: ( LE c,a,P & LE a,b,P & LE b,d,P ) ; :: thesis: c = d
LE a,c,P by A6, JORDAN6:58;
hence c = d by A2, A7, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A8: ( LE b,c,P & LE c,d,P & LE d,a,P ) ; :: thesis: c = d
then A9: LE c,a,P by JORDAN6:58;
thus c = d :: thesis: verum
proof
per cases ( ( LE a,b,P & LE b,d,P & LE d,c,P ) or ( LE b,d,P & LE d,c,P & LE c,a,P ) or ( LE d,c,P & LE c,a,P & LE a,b,P ) or ( LE c,a,P & LE a,b,P & LE b,d,P ) ) by A5;
suppose ( LE a,b,P & LE b,d,P & LE d,c,P ) ; :: thesis: c = d
hence c = d by A8, JORDAN6:57; :: thesis: verum
end;
suppose ( LE b,d,P & LE d,c,P & LE c,a,P ) ; :: thesis: c = d
hence c = d by A8, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,c,P & LE c,a,P & LE a,b,P ) ; :: thesis: c = d
hence c = d by A8, JORDAN6:57; :: thesis: verum
end;
suppose A10: ( LE c,a,P & LE a,b,P & LE b,d,P ) ; :: thesis: c = d
LE b,a,P by A8, A9, JORDAN6:58;
hence c = d by A1, A10, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A11: ( LE c,d,P & LE d,a,P & LE a,b,P ) ; :: thesis: c = d
then A12: LE d,b,P by JORDAN6:58;
thus c = d :: thesis: verum
proof
per cases ( ( LE a,b,P & LE b,d,P & LE d,c,P ) or ( LE b,d,P & LE d,c,P & LE c,a,P ) or ( LE d,c,P & LE c,a,P & LE a,b,P ) or ( LE c,a,P & LE a,b,P & LE b,d,P ) ) by A5;
suppose ( LE a,b,P & LE b,d,P & LE d,c,P ) ; :: thesis: c = d
hence c = d by A11, JORDAN6:57; :: thesis: verum
end;
suppose ( LE b,d,P & LE d,c,P & LE c,a,P ) ; :: thesis: c = d
hence c = d by A11, JORDAN6:57; :: thesis: verum
end;
suppose ( LE d,c,P & LE c,a,P & LE a,b,P ) ; :: thesis: c = d
hence c = d by A11, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,a,P & LE a,b,P & LE b,d,P ) ; :: thesis: c = d
hence c = d by A3, A12, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
suppose A13: ( LE d,a,P & LE a,b,P & LE b,c,P ) ; :: thesis: c = d
then A14: LE d,b,P by JORDAN6:58;
thus c = d :: thesis: verum
proof
per cases ( ( LE a,b,P & LE b,d,P & LE d,c,P ) or ( LE b,d,P & LE d,c,P & LE c,a,P ) or ( LE d,c,P & LE c,a,P & LE a,b,P ) or ( LE c,a,P & LE a,b,P & LE b,d,P ) ) by A5;
suppose ( LE a,b,P & LE b,d,P & LE d,c,P ) ; :: thesis: c = d
hence c = d by A3, A14, JORDAN6:57; :: thesis: verum
end;
suppose ( LE b,d,P & LE d,c,P & LE c,a,P ) ; :: thesis: c = d
hence c = d by A3, A14, JORDAN6:57; :: thesis: verum
end;
suppose A15: ( LE d,c,P & LE c,a,P & LE a,b,P ) ; :: thesis: c = d
LE a,c,P by A13, JORDAN6:58;
hence c = d by A2, A15, JORDAN6:57; :: thesis: verum
end;
suppose ( LE c,a,P & LE a,b,P & LE b,d,P ) ; :: thesis: c = d
hence c = d by A3, A14, JORDAN6:57; :: thesis: verum
end;
end;
end;
end;
end;