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

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

assume that
A1: d <> a 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 <> d & e <> a & a,b,d,e 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 and
A4: LE b,c,P and
A5: LE c,d,P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

thus ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P ) :: thesis: verum
proof
A6: LE b,d,P by ;
per cases ( a = W-min P or a <> W-min P ) ;
suppose A7: a = W-min P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

d in P by ;
then consider e being Point of () such that
A8: e <> d and
A9: LE d,e,P by Th7;
take e ; :: thesis: ( e <> d & e <> a & a,b,d,e are_in_this_order_on P )
thus e <> d by A8; :: thesis: ( e <> a & a,b,d,e are_in_this_order_on P )
thus e <> a by ; :: thesis: a,b,d,e are_in_this_order_on P
thus a,b,d,e are_in_this_order_on P by A3, A6, A9; :: thesis: verum
end;
suppose A10: a <> W-min P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

take e = W-min P; :: thesis: ( e <> d & e <> a & a,b,d,e are_in_this_order_on P )
a in P by ;
then A11: LE e,a,P by JORDAN7:3;
now :: thesis: not e = d
LE a,c,P by ;
then A12: LE a,d,P by ;
assume e = d ; :: thesis: contradiction
hence contradiction by A1, A11, A12, JORDAN6:57; :: thesis: verum
end;
hence e <> d ; :: thesis: ( e <> a & a,b,d,e are_in_this_order_on P )
thus e <> a by A10; :: thesis: a,b,d,e are_in_this_order_on P
thus a,b,d,e are_in_this_order_on P by A3, A6, A11; :: thesis: verum
end;
end;
end;
end;
suppose that A13: ( LE b,c,P & LE c,d,P ) and
A14: LE d,a,P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

consider e being Point of () such that
A15: ( e <> d & e <> a & LE d,e,P & LE e,a,P ) by A1, A14, Th8;
take e ; :: thesis: ( e <> d & e <> a & a,b,d,e are_in_this_order_on P )
LE b,d,P by ;
hence ( e <> d & e <> a & a,b,d,e are_in_this_order_on P ) by A15; :: thesis: verum
end;
suppose that LE c,d,P and
A16: LE d,a,P and
A17: LE a,b,P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

consider e being Point of () such that
A18: ( e <> d & e <> a & LE d,e,P & LE e,a,P ) by A1, A16, Th8;
take e ; :: thesis: ( e <> d & e <> a & a,b,d,e are_in_this_order_on P )
thus ( e <> d & e <> a & a,b,d,e are_in_this_order_on P ) by ; :: thesis: verum
end;
suppose that A19: LE d,a,P and
A20: LE a,b,P and
LE b,c,P ; :: thesis: ex e being Point of () st
( e <> d & e <> a & a,b,d,e are_in_this_order_on P )

consider e being Point of () such that
A21: ( e <> d & e <> a & LE d,e,P & LE e,a,P ) by A1, A19, Th8;
take e ; :: thesis: ( e <> d & e <> a & a,b,d,e are_in_this_order_on P )
thus ( e <> d & e <> a & a,b,d,e are_in_this_order_on P ) by ; :: thesis: verum
end;
end;