let FdSp be FanodesSp; :: thesis: for a, b, c, d being Element of FdSp st a <> b & a,b,c is_collinear & a,b,d is_collinear holds
a,c,d is_collinear
let a, b, c, d be Element of FdSp; :: thesis: ( a <> b & a,b,c is_collinear & a,b,d is_collinear implies a,c,d is_collinear )
assume that
A1: a <> b and
A2: ( a,b,c is_collinear & a,b,d is_collinear ) ; :: thesis: a,c,d is_collinear
( a,b '||' a,c & a,b '||' a,d ) by A2, Def2;
then a,c '||' a,d by A1, PARSP_1:def 12;
hence a,c,d is_collinear by Def2; :: thesis: verum