reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;
reserve G for IncProjectivePlane;
reserve a,q for POINT of G;
reserve A,B for LINE of G;
reserve G for IncProjSp;
reserve a,b,c,d for POINT of G;
reserve P for LINE of G;
reserve G for IncProjectivePlane;

theorem
  for G being IncProjectivePlane, q1,q2,r1,r2 being POINT of G, H being
  LINE of G st r1<>r2 & r1 on H & r2 on H & q1|'H & q2|'H holds q1*r1<>q2*r2
proof
  let G be IncProjectivePlane, q1,q2,r1,r2 be POINT of G, H be LINE of G such
  that
A1: r1<>r2 and
A2: r1 on H and
A3: r2 on H and
A4: q1|'H and
A5: q2|'H and
A6: q1*r1 = q2*r2;
  set L1=q1*r1,L2=q2*r2;
A7: q1 on L1 by A2,A4,Th16;
  r2 on L2 by A3,A5,Th16;
  then
A8: r2 on L2,H by A3;
  r1 on L1 by A2,A4,Th16;
  then r1 on L1,H by A2;
  then r1=L1*H by A4,A7,Def9;
  hence contradiction by A1,A4,A6,A7,A8,Def9;
end;
