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 Th15:
  not a on A & not a on C & not b on B & not b on C & not b on Q &
not A,B,C are_concurrent & a<>b & b<>q & A<>Q & {c,o} on A & {o,o99,d} on B & {
c,d,o9} on C & {a,b,d} on O & {c,oo9} on Q & {a,o,o9} on O1 & {b,o9,oo9} on O2
& {o,oo9,q} on O3 & q on O 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 a on C and
A3: not b on B and
A4: not b on C and
A5: not b on Q and
A6: not A,B,C are_concurrent and
A7: a<>b and
A8: b<>q and
A9: A<>Q and
A10: {c,o} on A and
A11: {o,o99,d} on B and
A12: {c,d,o9} on C and
A13: {a,b,d} on O and
A14: {c,oo9} on Q and
A15: {a,o,o9} on O1 and
A16: {b,o9,oo9} on O2 and
A17: {o,oo9,q} on O3 and
A18: q on O;
A19: o on A by A10,INCSP_1:1;
A20: c on A & c on Q by A10,A14,INCSP_1:1;
A21: o9 on C by A12,INCSP_1:2;
A22: oo9 on O2 by A16,INCSP_1:2;
A23: b on O by A13,INCSP_1:2;
A24: q on O3 by A17,INCSP_1:2;
A25: o on O3 & oo9 on O3 by A17,INCSP_1:2;
  set X = CHAIN(A);
  set f=IncProj(A,a,C),g=IncProj(C,b,B),f1=IncProj(A,q,Q),g1=IncProj (Q,b,B);
A26: o on B by A11,INCSP_1:2;
A27: dom f= CHAIN(A) by A1,A2,Th4;
A28: o9 on O2 by A16,INCSP_1:2;
A29: q on O3 by A17,INCSP_1:2;
A30: b on O2 by A16,INCSP_1:2;
A31: o on B by A11,INCSP_1:2;
A32: d on C by A12,INCSP_1:2;
  then
A33: o<>d by A6,A19,A31;
A34: d on O by A13,INCSP_1:2;
A35: o on O3 & oo9 on O3 by A17,INCSP_1:2;
A36: o9 on O1 by A15,INCSP_1:2;
A37: o on A by A10,INCSP_1:1;
A38: a on O1 by A15,INCSP_1:2;
A39: d on B by A11,INCSP_1:2;
  then
A40: q<>o by A3,A18,A31,A23,A34,A33,INCPROJ:def 4;
A41: c on C by A12,INCSP_1:2;
A42: oo9 on Q by A14,INCSP_1:1;
A43: o on O1 by A15,INCSP_1:2;
A44: c on A by A10,INCSP_1:1;
  then
A45: o<>c by A6,A31,A41;
  then
A46: o9<>c by A1,A44,A19,A38,A43,A36,INCPROJ:def 4;
  then
A47: c <>oo9 by A4,A41,A21,A30,A28,A22,INCPROJ:def 4;
A48: not q on A
  proof
    assume not thesis;
    then oo9 on A by A37,A35,A29,A40,INCPROJ:def 4;
    hence contradiction by A9,A20,A42,A47,INCPROJ:def 4;
  end;
A49: a on O by A13,INCSP_1:2;
  o9<>d
  proof
    assume not thesis;
    then O1=O by A2,A32,A49,A34,A38,A36,INCPROJ:def 4;
    hence contradiction by A3,A31,A39,A23,A34,A43,A33,INCPROJ:def 4;
  end;
  then O<>O2 by A2,A32,A21,A49,A34,A28,INCPROJ:def 4;
  then
A50: q<>oo9 by A8,A18,A23,A30,A22,INCPROJ:def 4;
A51: not q on Q
  proof
    assume not thesis;
    then o on Q by A42,A35,A29,A50,INCPROJ:def 4;
    hence contradiction by A9,A20,A37,A45,INCPROJ:def 4;
  end;
  then
A52: dom f1 = CHAIN(A) by A48,Th4;
  then
A53: dom (g1*f1) = X by A3,A5,A48,A51,PROJRED1:22;
A54: d on B & d on O by A11,A13,INCSP_1:2;
A55: O1<>O2
  proof
    assume not thesis;
    then o on O by A7,A49,A23,A38,A43,A30,INCPROJ:def 4;
    then O=B by A54,A26,A33,INCPROJ:def 4;
    hence contradiction by A3,A13,INCSP_1:2;
  end;
A56: c on Q & oo9 on Q by A14,INCSP_1:1;
A57: 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
A58: {o9,b,oo9} on O2 & {o9,o,a} on O1 by A38,A43,A36,A30,A28,A22,INCSP_1:2;
A59: {o,q,oo9} on O3 & {b,q,a} on O by A18,A49,A23,A25,A24,INCSP_1:2;
A60: O2,O1,C are_mutually_distinct by A2,A4,A38,A30,A55,ZFMISC_1:def 5;
    let x such that
A61: x on A;
A62: {c,o,x} on A by A44,A19,A61,INCSP_1:2;
A63: x in dom f1 by A52,A61;
A64: x in dom f by A27,A61;
    consider Q1 such that
A65: a on Q1 & x on Q1 by INCPROJ:def 5;
    consider x9 being POINT of IPP such that
A66: x9 on Q1 and
A67: x9 on C by INCPROJ:def 9;
A68: {x,a,x9} on Q1 by A65,A66,INCSP_1:2;
A69: {o9,c,x9} on C by A41,A21,A67,INCSP_1:2;
    consider Q2 such that
A70: x9 on Q2 and
A71: b on Q2 by INCPROJ:def 5;
    consider y such that
A72: y on Q and
A73: y on Q2 by INCPROJ:def 9;
A74: {c,y,oo9} on Q by A56,A72,INCSP_1:2;
    {b,y,x9} on Q2 by A70,A71,A73,INCSP_1:2;
    then consider R such that
A75: {y,q,x} on R by A1,A2,A4,A19,A21,A46,A58,A69,A62,A74,A68,A59,A60,
PROJRED1:12;
A76: x on R by A75,INCSP_1:2;
    y on R & q on R by A75,INCSP_1:2;
    then
A77: f1.x = y by A48,A51,A61,A72,A76,PROJRED1:def 1;
    consider x99 being POINT of IPP such that
A78: x99 on Q2 & x99 on B by INCPROJ:def 9;
A79: g1.y = x99 by A3,A5,A71,A78,A72,A73,PROJRED1:def 1;
A80: g.x9 = x99 by A3,A4,A67,A70,A71,A78,PROJRED1:def 1;
    f.x = x9 by A1,A2,A61,A65,A66,A67,PROJRED1:def 1;
    then (g*f).x = g1.(f1.x) by A80,A77,A79,A64,FUNCT_1:13
      .= (g1*f1).x by A63,FUNCT_1:13;
    hence thesis;
  end;
A81: 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 A57;
  end;
  dom (g*f) = X by A1,A2,A3,A4,A27,PROJRED1:22;
  hence thesis by A53,A81,FUNCT_1:2;
end;
