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 Th23:
  not a on A & not b on B & not a on C & not b on C & a<>b & a on
O & b on O & q on O & not q on A & q<>b & not A,B,C are_concurrent implies ex Q
st A,C,Q are_concurrent & not b on Q & not q on Q & IncProj(C,b,B)*IncProj(A,a,
  C) = IncProj(Q,b,B)* IncProj(A,q,Q)
proof
  consider c such that
A1: c on A & c on C by INCPROJ:def 9;
  assume
A2: ( not a on A)& not b on B & ( not a on C)& not b on C & a<>b & a on
  O & b on O & q on O & ( not q on A)& q<>b & not A,B,C are_concurrent;
A3: now
    assume B,C,O are_concurrent;
    then consider Q such that
A4: c on Q and
A5: ( not b on Q)& not q on Q & IncProj(C,b,B)*IncProj(A,a,C) =
    IncProj(Q,b,B) *IncProj(A,q,Q) by A2,A1,Lm4;
    take Q;
    thus A,C,Q are_concurrent by A1,A4;
    thus not b on Q & not q on Q & IncProj(C,b,B)*IncProj(A,a,C) = IncProj(Q,b
    ,B) * IncProj(A,q,Q) by A5;
  end;
  now
    assume not B,C,O are_concurrent;
    then consider Q such that
A6: c on Q and
A7: ( not b on Q)& not q on Q & IncProj(C,b,B)*IncProj(A,a,C) =
    IncProj(Q,b,B) *IncProj(A,q,Q) by A2,A1,Lm3;
    take Q;
    thus A,C,Q are_concurrent by A1,A6;
    thus not b on Q & not q on Q & IncProj(C,b,B)*IncProj(A,a,C) = IncProj(Q,b
    ,B)* IncProj(A,q,Q) by A7;
  end;
  hence thesis by A3;
end;
