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 o on A & not o on B & not o on C implies IncProj(C,o,B)*IncProj(A,
  o,C) = IncProj(A,o,B)
proof
  assume that
A1: not o on A and
A2: not o on B and
A3: not o on C;
  set f=IncProj(A,o,B),g=IncProj(C,o,B),h=IncProj(A,o,C);
A4: dom f= CHAIN(A) by A1,A2,Th4;
  set X = CHAIN(A);
A5: dom h = CHAIN(A) by A1,A3,Th4;
A6: for x st x on A holds IncProj(A,o,B).x = (IncProj(C,o,B)*IncProj(A,o,C)) .x
  proof
    let x such that
A7: x on A;
    consider Q1 such that
A8: o on Q1 and
A9: x on Q1 by INCPROJ:def 5;
    consider x9 being POINT of IPP such that
A10: x9 on Q1 & x9 on C by INCPROJ:def 9;
A11: h.x = x9 by A1,A3,A7,A8,A9,A10,PROJRED1:def 1;
    consider y being POINT of IPP such that
A12: y on Q1 & y on B by INCPROJ:def 9;
A13: f.x = y by A1,A2,A7,A8,A9,A12,PROJRED1:def 1;
    x in dom h by A5,A7;
    then (g*h).x = g.(h.x) by FUNCT_1:13
      .= f.x by A2,A3,A8,A10,A12,A13,A11,PROJRED1:def 1;
    hence thesis;
  end;
A14: now
    let y be object;
    assume y in X;
    then ex x st y=x & x on A;
    hence (g*h).y = f.y by A6;
  end;
  dom (g*h) = X by A1,A2,A3,A5,PROJRED1:22;
  hence thesis by A4,A14,FUNCT_1:2;
end;
