theorem
  for e,x1,x2,p1,p2 being POINT of G for H,I being LINE of G st x1 on I
& x2 on I & e |'I & x1<>x2 & p1<>e & p2 <>e & p1 on e*x1 & p2 on e*x2 holds ex
  r being POINT of G st r on p1*p2 & r on H & r<>e
proof
  let e,x1,x2,p1,p2 be POINT of G;
  let H,I be LINE of G such that
A1: x1 on I and
A2: x2 on I and
A3: e|'I and
A4: x1<>x2 and
A5: p1<>e and
A6: p2<>e and
A7: p1 on e*x1 and
A8: p2 on e*x2;
  set L1=e*x1,L2=e*x2,L=p1*p2;
A9: e on L1 by A1,A3,Th16;
A10: e on L2 by A2,A3,Th16;
  x1 on L1 & x2 on L2 by A1,A2,A3,Th16;
  then
A11: L1 <> L2 by A1,A2,A3,A4,A9,INCPROJ:def 4;
  then
A12: p1<>p2 by A5,A7,A8,A9,A10,INCPROJ:def 4;
  then p2 on L by Th16;
  then
A13: L<>L1 by A6,A8,A9,A10,A11,INCPROJ:def 4;
  consider r being POINT of G such that
A14: r on L and
A15: r on H by INCPROJ:def 9;
  p1 on L by A12,Th16;
  then r<>e by A5,A7,A14,A9,A13,INCPROJ:def 4;
  hence thesis by A14,A15;
end;
