let PCPP be CollProjectiveSpace; for p1, p2, p3, q1, q2 being Element of PCPP st not p1,p2,q1 are_collinear & p1,p2,p3 are_collinear & q1,q2,p3 are_collinear & p3 <> q2 & p2 <> p3 holds
not p3,p2,q2 are_collinear
let p1, p2, p3, q1, q2 be Element of PCPP; ( not p1,p2,q1 are_collinear & p1,p2,p3 are_collinear & q1,q2,p3 are_collinear & p3 <> q2 & p2 <> p3 implies not p3,p2,q2 are_collinear )
assume that
A1:
not p1,p2,q1 are_collinear
and
A2:
p1,p2,p3 are_collinear
and
A3:
q1,q2,p3 are_collinear
and
A4:
p3 <> q2
and
A5:
p2 <> p3
; not p3,p2,q2 are_collinear
assume A6:
p3,p2,q2 are_collinear
; contradiction
p3,p2,p1 are_collinear
by A2, Th1;
then A7:
p3,q2,p1 are_collinear
by A5, A6, Th2;
A8:
p3,q2,q1 are_collinear
by A3, Th1;
p3,q2,p2 are_collinear
by A6, Th1;
hence
contradiction
by A1, A4, A7, A8, ANPROJ_2:def 8; verum