let FdSp be FanodesSp; :: thesis: for a, b, c, d being Element of FdSp st a <> b & a,b,c are_collinear & a,b '||' c,d holds
c,b '||' c,d
let a, b, c, d be Element of FdSp; :: thesis: ( a <> b & a,b,c are_collinear & a,b '||' c,d implies c,b '||' c,d )
assume that
A1: a <> b and
A2: a,b,c are_collinear and
A3: a,b '||' c,d ; :: thesis: c,b '||' c,d

now :: thesis: ( a <> c implies c,b '||' c,d )

a,b '||' a,c by A2;
then A4: ( a,c '||' c,b & a,c '||' c,d ) by A1, A3, PARSP_1:24, PARSP_1:def 12;
assume a <> c ; :: thesis: c,b '||' c,d
hence c,b '||' c,d by A4, PARSP_1:def 12; :: thesis: verum

end;

hence c,b '||' c,d by A3; :: thesis: verum