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
  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 B & q<>a & not A,B,C are_concurrent implies ex Q st B,
C,Q are_concurrent & not a on Q & not q on Q & IncProj(C,b,B)*IncProj(A,a,C) =
  IncProj(Q,q,B)* IncProj(A,a,Q)
proof
  assume that
A1: not a on A and
A2: not b on B and
A3: not a on C and
A4: not b on C and
A5: a<>b & a on O & b on O & q on O and
A6: not q on B and
A7: q<>a and
A8: not A,B,C are_concurrent;
A9: IncProj(C,a,A) is one-to-one & IncProj(B,b,C) is one-to-one by A1,A2,A3,A4
,Th7;
  not B,A,C are_concurrent
  by A8;
  then consider Q such that
A10: B,C,Q are_concurrent and
A11: not a on Q and
A12: not q on Q and
A13: IncProj(C,a,A)*IncProj(B,b,C) = IncProj(Q,a,A)*IncProj (B,q,Q) by A1,A2,A3
,A4,A5,A6,A7,Th23;
A14: IncProj(Q,a,A) is one-to-one & IncProj(B,q,Q) is one-to-one by A1,A6,A11
,A12,Th7;
  take Q;
  thus B,C,Q are_concurrent & not a on Q & not q on Q by A10,A11,A12;
  thus IncProj(C,b,B)*IncProj(A,a,C) = IncProj(B,b,C)"*IncProj (A,a,C) by A2,A4
,Th8
    .= IncProj(B,b,C)"*IncProj(C,a,A)" by A1,A3,Th8
    .= (IncProj(Q,a,A)*IncProj (B,q,Q))" by A13,A9,FUNCT_1:44
    .= IncProj(B,q,Q)"*IncProj(Q,a,A)" by A14,FUNCT_1:44
    .= IncProj(B,q,Q)"*IncProj(A,a,Q) by A1,A11,Th8
    .= IncProj(Q,q,B)*IncProj (A,a,Q) by A6,A12,Th8;
end;
