let AFS be AffinSpace; :: thesis: for a, b, c, d being Element of AFS holds
( not a,b // c,d or a,c // b,d or ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
let a, b, c, d be Element of AFS; :: thesis: ( not a,b // c,d or a,c // b,d or ex x being Element of AFS 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 AFS st
( LIN a,c,x & LIN b,d,x ) )
assume A2: not a,c // b,d ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )

A3: now

assume a = b ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
then ( LIN a,c,a & LIN b,d,a ) by AFF_1:16;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) ; :: thesis: verum

end;

now

assume A4: a <> b ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
consider z being Element of AFS such that
A5: ( a,b // c,z & a,c // b,z ) by DIRAF:47;

A6: now

assume b = z ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
then b,a // b,c by A5, AFF_1:13;
then LIN b,a,c by AFF_1:def 1;
then ( LIN a,c,b & LIN b,d,b ) by AFF_1:15, AFF_1:16;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) ; :: thesis: verum

end;

now

assume A7: b <> z ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
c,d // c,z by A1, A4, A5, AFF_1:14;
then LIN c,d,z by AFF_1:def 1;
then LIN d,c,z by AFF_1:15;
then d,c // d,z by AFF_1:def 1;
then z,d // d,c by AFF_1:13;
then consider t being Element of AFS such that
A8: ( b,d // d,t & b,z // c,t ) by A2, A5, DIRAF:47;
a,c // c,t by A5, A7, A8, AFF_1:14;
then ( c,a // c,t & d,b // d,t ) by A8, AFF_1:13;
then ( LIN c,a,t & LIN d,b,t ) by AFF_1:def 1;
then ( LIN a,c,t & LIN b,d,t ) by AFF_1:15;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) ; :: thesis: verum

end;

hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) by A6; :: thesis: verum

end;

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