let FdSp be FanodesSp; :: thesis: for a, b, c, d being Element of FdSp st a <> b & a,b,c is_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 is_collinear & a,b '||' c,d implies c,b '||' c,d )
assume A1: ( a <> b & a,b,c is_collinear & a,b '||' c,d ) ; :: thesis: c,b '||' c,d

now

assume a <> c ; :: thesis: c,b '||' c,d
then ( a <> c & a,b '||' a,c ) by A1, Def2;
then ( a <> c & a,c '||' c,b & a,c '||' c,d ) by A1, PARSP_1:41, PARSP_1:def 12;
hence c,b '||' c,d by PARSP_1:def 12; :: thesis: verum

end;

hence c,b '||' c,d by A1; :: thesis: verum