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
( a,c,x are_collinear & b,d,x are_collinear ) )

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
( a,c,x are_collinear & b,d,x are_collinear ) )

assume A1: a,b '||' c,d ; :: thesis: ( a,c '||' b,d or ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) )

assume A2: not a,c '||' b,d ; :: thesis: ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear )

A3: now :: thesis: ( a <> b implies ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) )
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
( a,c,x are_collinear & b,d,x are_collinear )

A7: now :: thesis: ( b <> z implies ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) )
end;
now :: thesis: ( b = z implies ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) )
end;
hence ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) by A7; :: thesis: verum
end;
now :: thesis: ( a = b implies ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) )
assume a = b ; :: thesis: ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear )

then ( a,c,a are_collinear & b,d,a are_collinear ) by DIRAF:31;
hence ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) ; :: thesis: verum
end;
hence ex x being Element of OAS st
( a,c,x are_collinear & b,d,x are_collinear ) by A3; :: thesis: verum