let PCPP be CollProjectiveSpace; :: thesis: 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; :: thesis: ( 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 ; :: thesis: not p3,p2,q2 are_collinear
assume A6: p3,p2,q2 are_collinear ; :: thesis: 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; :: thesis: verum