let FdSp be FanodesSp; :: thesis: for a, b, p, q, r being Element of FdSp st a <> b & a,b,p are_collinear & a,b,q are_collinear & a,b,r are_collinear holds
p,q,r are_collinear
let a, b, p, q, r be Element of FdSp; :: thesis: ( a <> b & a,b,p are_collinear & a,b,q are_collinear & a,b,r are_collinear implies p,q,r are_collinear )
assume that
A1: a <> b and
A2: a,b,p are_collinear and
A3: a,b,q are_collinear and
A4: a,b,r are_collinear ; :: thesis: p,q,r are_collinear
A5: a,b '||' a,p by A2;
a,b '||' a,r by A4;
then A6: a,b '||' p,r by A5, PARSP_1:35;
a,b '||' a,q by A3;
then a,b '||' p,q by A5, PARSP_1:35;
then p,q '||' p,r by A1, A6, PARSP_1:def 12;
hence p,q,r are_collinear ; :: thesis: verum