let FdSp be FanodesSp; :: thesis: for a, b, p, q, r being Element of FdSp st a <> b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear holds
p,q,r is_collinear
let a, b, p, q, r be Element of FdSp; :: thesis: ( a <> b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear implies p,q,r is_collinear )
assume ( a <> b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear ) ; :: thesis: p,q,r is_collinear
then ( a,b '||' a,p & a,b '||' a,q & a,b '||' a,r & a <> b ) by Def2;
then ( a <> b & a,b '||' p,q & a,b '||' p,r ) by PARSP_1:53;
then p,q '||' p,r by PARSP_1:def 12;
hence p,q,r is_collinear by Def2; :: thesis: verum