reserve IPS for IncProjSp,
  z for POINT of IPS;
reserve IPP for Desarguesian 2-dimensional IncProjSp,
  a,b,c,d,p,pp9,q,o,o9,o99 ,oo9 for POINT of IPP,
  r,s,x,y,o1,o2 for POINT of IPP,
  O1,O2,O3,O4,A,B,C,O,Q,Q1 ,Q2,Q3,R,S,X for LINE of IPP;

theorem Th20:
  not a on A & not b on B & not a on C & not b on C & not A,B,C
  are_concurrent & A,C,Q are_concurrent & not b on Q & A<>Q & a<>b & a on O & b
on O implies ex q st q on O & not q on A & not q on Q & IncProj(C,b,B)*IncProj(
  A,a,C) = IncProj(Q,b,B)*IncProj(A,q,Q)
proof
  assume
A1: ( not a on A)& not b on B & ( not a on C)& not b on C & ( not A,B,C
are_concurrent)& A,C,Q are_concurrent & ( not b on Q)& A<>Q & a<>b & a on O & b
  on O;
  then not B,C,O are_concurrent implies thesis by Lm1;
  hence thesis by A1,Lm2;
end;
