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 & not a on B & not b
on A & c on A & c on C & d on B & d on C & a on S & d on S & c on R & b on R &
s on A & s on S & r on B & r on R & s on Q & r on Q & not A,B,C are_concurrent
  implies IncProj(C,b,B)*IncProj(A,a,C) = IncProj(Q,a,B)*IncProj(A,b,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 on B and
A6: not b on A and
A7: c on A and
A8: c on C and
A9: d on B and
A10: d on C and
A11: a on S and
A12: d on S and
A13: c on R and
A14: b on R and
A15: s on A and
A16: s on S and
A17: r on B and
A18: r on R and
A19: s on Q and
A20: r on Q and
A21: not A,B,C are_concurrent;
A22: c <>d by A7,A8,A9,A21;
  then
A23: c <>s by A3,A8,A10,A11,A12,A16,INCPROJ:def 4;
A24: now
    assume b on Q;
    then R=Q by A2,A14,A17,A18,A20,INCPROJ:def 4;
    hence contradiction by A6,A7,A13,A14,A15,A19,A23,INCPROJ:def 4;
  end;
A25: d<>r by A4,A8,A10,A13,A14,A18,A22,INCPROJ:def 4;
A26: now
    assume a on Q;
    then S=Q by A1,A11,A15,A16,A19,INCPROJ:def 4;
    hence contradiction by A5,A9,A11,A12,A17,A20,A25,INCPROJ:def 4;
  end;
  set X = CHAIN(A);
  set f=IncProj(A,a,C),g=IncProj(C,b,B),g1=IncProj(Q,a,B),f1=IncProj(A,b,Q);
A27: dom f= CHAIN(A) by A1,A3,Th4;
A28: dom f1 = CHAIN(A) by A6,A24,Th4;
  then
A29: dom (g1*f1) = X by A5,A6,A26,A24,PROJRED1:22;
  A<>C by A21,Th1;
  then
A30: C,A,R are_mutually_distinct by A4,A6,A14,ZFMISC_1:def 5;
A31: c <>d by A7,A8,A9,A21;
A32: for x st x on A holds (IncProj(C,b,B)*IncProj(A,a,C)).x = (IncProj(Q,a,
  B)* IncProj(A,b,Q)).x
  proof
    let x;
    assume
A33: x on A;
    then reconsider x9=f.x,y=f1.x as POINT of IPP by A1,A3,A6,A24,PROJRED1:19;
A34: x in dom f1 by A28,A33;
A35: x9 on C by A1,A3,A33,PROJRED1:20;
    then reconsider x99=g.x9 as POINT of IPP by A2,A4,PROJRED1:19;
    consider O1 such that
A36: a on O1 & x on O1 & x9 on O1 by A1,A3,A33,A35,PROJRED1:def 1;
A37: y on Q by A6,A24,A33,PROJRED1:20;
    then consider O3 such that
A38: b on O3 & x on O3 & y on O3 by A6,A24,A33,PROJRED1:def 1;
A39: x99 on B by A2,A4,A35,PROJRED1:20;
    then consider O2 such that
A40: b on O2 & x9 on O2 & x99 on O2 by A2,A4,A35,PROJRED1:def 1;
A41: now
A42:  {y,s,r} on Q & {d,x99,r} on B by A9,A17,A19,A20,A37,A39,INCSP_1:2;
      assume
A43:  s<>x;
A44:  {d,a,s} on S by A11,A12,A16,INCSP_1:2;
A45:  {c,b,r} on R & {b,x,y} on O3 by A13,A14,A18,A38,INCSP_1:2;
A46:  {b,x99,x9} on O2 & {x,a,x9} on O1 by A36,A40,INCSP_1:2;
      {c,d,x9} on C & {c,x,s} on A by A7,A8,A10,A15,A33,A35,INCSP_1:2;
      then consider O4 such that
A47:  {x99,a,y} on O4 by A6,A7,A31,A23,A30,A43,A45,A46,A42,A44,PROJRED1:12;
A48:  y on O4 by A47,INCSP_1:2;
      x99 on O4 & a on O4 by A47,INCSP_1:2;
      hence g.(f.x) = g1.(f1.x) by A5,A26,A37,A39,A48,PROJRED1:def 1;
    end;
A49: now
      assume
A50:  s=x;
      then x9=d by A1,A3,A10,A11,A12,A15,A16,PROJRED1:def 1;
      then
A51:  x99=d by A2,A4,A9,A10,PROJRED1:24;
      y=s by A6,A15,A19,A24,A50,PROJRED1:24;
      hence g.(f.x) = g1.(f1.x) by A5,A9,A11,A12,A16,A19,A26,A51,PROJRED1:def 1
;
    end;
    x in dom f by A27,A33;
    then (g*f).x = g1.(f1.x) by A49,A41,FUNCT_1:13
      .= (g1*f1).x by A34,FUNCT_1:13;
    hence thesis;
  end;
A52: 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 A32;
  end;
  dom (g*f) = X by A1,A2,A3,A4,A27,PROJRED1:22;
  hence thesis by A29,A52,FUNCT_1:2;
end;
