let P be Simple_closed_curve; :: thesis: for a, b, c, d being Point of () st c <> d & a,b,c,d are_in_this_order_on P holds
ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

let a, b, c, d be Point of (); :: thesis: ( c <> d & a,b,c,d are_in_this_order_on P implies ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P ) )

assume that
A1: c <> d and
A2: ( ( 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 ) ) ; :: according to JORDAN17:def 1 :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

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 A2;
suppose that A3: ( LE a,b,P & LE b,c,P ) and
A4: LE c,d,P ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

consider e being Point of () such that
A5: ( e <> c & e <> d & LE c,e,P & LE e,d,P ) by A1, A4, Th8;
take e ; :: thesis: ( e <> c & e <> d & b,c,e,d are_in_this_order_on P )
thus ( e <> c & e <> d & b,c,e,d are_in_this_order_on P ) by A3, A5; :: thesis: verum
end;
suppose that A6: LE b,c,P and
A7: LE c,d,P and
LE d,a,P ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

consider e being Point of () such that
A8: ( e <> c & e <> d & LE c,e,P & LE e,d,P ) by A1, A7, Th8;
take e ; :: thesis: ( e <> c & e <> d & b,c,e,d are_in_this_order_on P )
thus ( e <> c & e <> d & b,c,e,d are_in_this_order_on P ) by A6, A8; :: thesis: verum
end;
suppose that A9: LE c,d,P and
A10: ( LE d,a,P & LE a,b,P ) ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

consider e being Point of () such that
A11: ( e <> c & e <> d & LE c,e,P & LE e,d,P ) by A1, A9, Th8;
take e ; :: thesis: ( e <> c & e <> d & b,c,e,d are_in_this_order_on P )
LE d,b,P by ;
hence ( e <> c & e <> d & b,c,e,d are_in_this_order_on P ) by A11; :: thesis: verum
end;
suppose that A12: LE d,a,P and
A13: LE a,b,P and
A14: LE b,c,P ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

thus ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P ) :: thesis: verum
proof
A15: LE d,b,P by ;
per cases ( d = W-min P or d <> W-min P ) ;
suppose A16: d = W-min P ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

c in P by ;
then consider e being Point of () such that
A17: e <> c and
A18: LE c,e,P by Th7;
take e ; :: thesis: ( e <> c & e <> d & b,c,e,d are_in_this_order_on P )
thus e <> c by A17; :: thesis: ( e <> d & b,c,e,d are_in_this_order_on P )
thus e <> d by ; :: thesis: b,c,e,d are_in_this_order_on P
thus b,c,e,d are_in_this_order_on P by ; :: thesis: verum
end;
suppose A19: d <> W-min P ; :: thesis: ex e being Point of () st
( e <> c & e <> d & b,c,e,d are_in_this_order_on P )

take e = W-min P; :: thesis: ( e <> c & e <> d & b,c,e,d are_in_this_order_on P )
d in P by ;
then A20: LE e,d,P by JORDAN7:3;
now :: thesis: not e = c
assume A21: e = c ; :: thesis: contradiction
LE d,c,P by ;
hence contradiction by A1, A20, A21, JORDAN6:57; :: thesis: verum
end;
hence e <> c ; :: thesis: ( e <> d & b,c,e,d are_in_this_order_on P )
thus e <> d by A19; :: thesis: b,c,e,d are_in_this_order_on P
thus b,c,e,d are_in_this_order_on P by ; :: thesis: verum
end;
end;
end;
end;
end;