reserve IPP for IncProjSp;
reserve a,b,c,d,p,q,o,r,s,t,u,v,w,x,y for POINT of IPP;
reserve A,B,C,D,L,P,Q,S for LINE of IPP;
reserve IPP for Fanoian IncProjSp;
reserve a,b,c,d,p,q,r,s for POINT of IPP;
reserve A,B,C,D,L,Q,R,S for LINE of IPP;
reserve IPP for Desarguesian 2-dimensional IncProjSp;
reserve c,p,q,x,y for POINT of IPP;
reserve A,B,C,D,K,L,R,X for LINE of IPP;
reserve f for PartFunc of the Points of IPP,the Points of IPP;

theorem Th22:
  not p on K & not p on L & not q on L & not q on R implies dom (
  IncProj(L,q,R)*IncProj(K,p,L)) = dom IncProj(K,p,L) & rng (IncProj(L,q,R)*
  IncProj(K,p,L)) = rng IncProj(L,q,R)
proof
  assume that
A1: ( not p on K)& not p on L and
A2: ( not q on L)& not q on R;
  for x being object holds x in dom (IncProj(L,q,R)*IncProj(K,p,L)) iff x in
  dom IncProj(K,p,L)
  proof
    let x be object;
    x in dom IncProj(K,p,L) implies x in dom (IncProj(L,q,R)*IncProj(K,p,L ))
    proof
      assume
A3:   x in dom IncProj(K,p,L);
      then reconsider x9 = x as POINT of IPP;
      consider A such that
A4:   x9 on A & p on A by INCPROJ:def 5;
      consider y such that
A5:   y on A and
A6:   y on L by INCPROJ:def 9;
A7:   y in dom IncProj(L,q,R) by A2,A6,Def1;
      x9 on K by A1,A3,Def1;
      then (IncProj(K,p,L)).x=y by A1,A4,A5,A6,Def1;
      hence thesis by A3,A7,FUNCT_1:11;
    end;
    hence thesis by FUNCT_1:11;
  end;
  hence dom (IncProj(L,q,R)*IncProj(K,p,L)) = dom IncProj(K,p,L) by TARSKI:2;
  for x being object st x in dom IncProj(L,q,R) holds x in rng IncProj(K,p,L)
  proof
    let x be object;
    assume
A8: x in dom IncProj(L,q,R);
    thus x in rng IncProj(K,p,L)
    proof
      reconsider x9 = x as POINT of IPP by A8;
A9:   x9 on L by A2,A8,Def1;
      ex y st y in dom IncProj(K,p,L) & x9 = (IncProj (K,p,L)).y
      proof
        consider A such that
A10:    x9 on A & p on A by INCPROJ:def 5;
        consider y being POINT of IPP such that
A11:    y on A and
A12:    y on K by INCPROJ:def 9;
        take y;
        thus y in dom IncProj(K,p,L) by A1,A12,Def1;
        thus thesis by A1,A9,A10,A11,A12,Def1;
      end;
      hence thesis by FUNCT_1:def 3;
    end;
  end;
  then dom IncProj(L,q,R) c= rng IncProj(K,p,L);
  hence rng (IncProj(L,q,R)*IncProj(K,p,L)) = rng IncProj(L,q,R) by RELAT_1:28;
end;
