let OAS be OAffinSpace; :: thesis: for a, b, c, d being Element of OAS holds
( not a,b '||' c,d or a,c '||' b,d or ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) )
let a, b, c, d be Element of OAS; :: thesis: ( not a,b '||' c,d or a,c '||' b,d or ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) )
assume A1: a,b '||' c,d ; :: thesis: ( a,c '||' b,d or ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) )
assume A2: not a,c '||' b,d ; :: thesis: ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x )

A3: now

consider z being Element of OAS such that
A4: a,b '||' c,z and
A5: a,c '||' b,z by DIRAF:26;
assume A6: a <> b ; :: thesis: ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x )

A7: now

c,d '||' c,z by A1, A6, A4, DIRAF:23;
then LIN c,d,z by DIRAF:def 5;
then LIN d,c,z by DIRAF:30;
then d,c '||' d,z by DIRAF:def 5;
then z,d '||' d,c by DIRAF:22;
then consider t being Element of OAS such that
A8: b,d '||' d,t and
A9: b,z '||' c,t by A2, A5, DIRAF:27;
assume b <> z ; :: thesis: ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x )
then a,c '||' c,t by A5, A9, DIRAF:23;
then c,a '||' c,t by DIRAF:22;
then LIN c,a,t by DIRAF:def 5;
then A10: LIN a,c,t by DIRAF:30;
d,b '||' d,t by A8, DIRAF:22;
then LIN d,b,t by DIRAF:def 5;
then LIN b,d,t by DIRAF:30;
hence ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) by A10; :: thesis: verum

end;

now

assume b = z ; :: thesis: ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x )
then b,a '||' b,c by A4, DIRAF:22;
then LIN b,a,c by DIRAF:def 5;
then A11: LIN a,c,b by DIRAF:30;
LIN b,d,b by DIRAF:31;
hence ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) by A11; :: thesis: verum

end;

hence ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) by A7; :: thesis: verum

end;

now

assume a = b ; :: thesis: ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x )
then ( LIN a,c,a & LIN b,d,a ) by DIRAF:31;
hence ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) ; :: thesis: verum

end;

hence ex x being Element of OAS st
( LIN a,c,x & LIN b,d,x ) by A3; :: thesis: verum