let FCPS be up-3-dimensional CollProjectiveSpace; for a, b, b9, c being Element of FCPS st not a,b,c are_collinear & a,b,b9 are_collinear & a <> b9 holds
not a,b9,c are_collinear
let a, b, b9, c be Element of FCPS; ( not a,b,c are_collinear & a,b,b9 are_collinear & a <> b9 implies not a,b9,c are_collinear )
assume that
A1:
not a,b,c are_collinear
and
A2:
a,b,b9 are_collinear
and
A3:
a <> b9
; not a,b9,c are_collinear
assume A4:
a,b9,c are_collinear
; contradiction
a,b9,b are_collinear
by A2, Th1;
hence
contradiction
by A1, A3, A4, COLLSP:6; verum