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 Th14:
  not a on A & not b on B & not a on C & not b on C & not A,B,C
are_concurrent & c on A & c on C & c on Q & not b on Q & A<>Q & a on O & b on O
  & not B,C,O are_concurrent & d on C & d on B & a on O1 & d on O1 & p on A & p
on O1 & q on O & q on O2 & p on O2 & pp9 on O2 & d on O3 & b on O3 & pp9 on O3
& pp9 on Q & q<>a & not q on A & not q on Q implies IncProj(C,b,B)*IncProj(A,a,
  C) = IncProj(Q,b,B)*IncProj(A,q,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: not A,B,C are_concurrent and
A6: c on A and
A7: c on C and
A8: c on Q and
A9: not b on Q and
A10: A<>Q and
A11: a on O and
A12: b on O and
A13: not B,C,O are_concurrent and
A14: d on C and
A15: d on B and
A16: a on O1 and
A17: d on O1 and
A18: p on A and
A19: p on O1 and
A20: q on O and
A21: q on O2 and
A22: p on O2 & pp9 on O2 and
A23: d on O3 & b on O3 and
A24: pp9 on O3 and
A25: pp9 on Q and
A26: q<>a and
A27: not q on A and
A28: not q on Q;
  set f=IncProj(A,a,C),g=IncProj(C,b,B),g1=IncProj(Q,b,B),f1=IncProj (A,q,Q);
A29: dom f= CHAIN(A) by A1,A3,Th4;
  set X = CHAIN(A);
A30: dom f1 = CHAIN(A) by A27,A28,Th4;
  then
A31: dom (g1*f1) = X by A2,A9,A27,A28,PROJRED1:22;
A32: O1<>O2
  proof
    assume not thesis;
    then d on O by A11,A16,A17,A20,A21,A26,INCPROJ:def 4;
    hence contradiction by A13,A14,A15;
  end;
  c <>d by A5,A6,A7,A15;
  then p<>c by A3,A7,A14,A16,A17,A19,INCPROJ:def 4;
  then
A33: pp9<>p by A6,A8,A10,A18,A25,INCPROJ:def 4;
A34: for x st x on A holds (IncProj(C,b,B)*IncProj(A,a,C)).x = (IncProj(Q,b,
  B)* IncProj(A,q,Q)).x
  proof
    let x such that
A35: x on A;
    consider Q3 such that
A36: q on Q3 & x on Q3 by INCPROJ:def 5;
    consider Q1 such that
A37: a on Q1 & x on Q1 by INCPROJ:def 5;
    consider y being POINT of IPP such that
A38: y on Q3 and
A39: y on Q by INCPROJ:def 9;
    consider x9 being POINT of IPP such that
A40: x9 on Q1 and
A41: x9 on C by INCPROJ:def 9;
A42: now
A43:  {pp9,y,c} on Q & {d,x9,c} on C by A7,A8,A14,A25,A41,A39,INCSP_1:2;
A44:  {p,pp9,q} on O2 & {pp9,d,b} on O3 by A21,A22,A23,A24,INCSP_1:2;
A45:  {b,a,q} on O & {x,y,q} on Q3 by A11,A12,A20,A36,A38,INCSP_1:2;
A46:  {p,x,c} on A & {p,d,a} on O1 by A6,A16,A17,A18,A19,A35,INCSP_1:2;
      assume
A47:  p<>x;
      {x,x9,a} on Q1 & A,O1,O2 are_mutually_distinct by A1,A16,A21,A27,A32,A37
,A40,INCSP_1:2,ZFMISC_1:def 5;
      then consider R such that
A48:  {y,x9,b} on R by A1,A3,A14,A18,A33,A47,A46,A44,A43,A45,PROJRED1:12;
A49:  b on R by A48,INCSP_1:2;
      consider x99 being POINT of IPP such that
A50:  x99 on R & x99 on B by INCPROJ:def 9;
      x9 on R by A48,INCSP_1:2;
      then
A51:  g.x9 = x99 by A2,A4,A41,A50,A49,PROJRED1:def 1;
A52:  x in dom f1 by A30,A35;
      y on R by A48,INCSP_1:2;
      then
A53:  g1.y = x99 by A2,A9,A39,A50,A49,PROJRED1:def 1;
A54:  x in dom f by A29,A35;
      f.x = x9 & f1.x = y by A1,A3,A27,A28,A35,A37,A40,A41,A36,A38,A39,
PROJRED1:def 1;
      then (g*f).x = g1.(f1.x) by A51,A53,A54,FUNCT_1:13
        .= (g1*f1).x by A52,FUNCT_1:13;
      hence thesis;
    end;
    now
A55:  f1.p=pp9 & g1.pp9=d by A2,A9,A15,A18,A21,A22,A23,A24,A25,A27,A28,
PROJRED1:def 1;
      assume
A56:  p=x;
A57:  x in dom f1 by A30,A35;
A58:  x in dom f by A29,A35;
      f.p=d & g.d=d by A1,A2,A3,A4,A14,A15,A16,A17,A18,A19,A23,PROJRED1:def 1;
      then (g*f).x = g1.(f1.x) by A56,A55,A58,FUNCT_1:13
        .= (g1*f1).x by A57,FUNCT_1:13;
      hence thesis;
    end;
    hence thesis by A42;
  end;
A59: now
    let y be object;
    assume y in X;
    then ex x st y=x & x on A;
    hence (g*f).y = (g1*f1).y by A34;
  end;
  dom (g*f) = X by A1,A2,A3,A4,A29,PROJRED1:22;
  hence thesis by A31,A59,FUNCT_1:2;
end;
