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;
reserve X for set;

theorem
  not p on K & not p on L & not q on L & not q on R & c on K & c on L &
  c on R & K <> R implies ex o being POINT of IPP st not o on K & not o on R &
  IncProj(L,q,R)*IncProj(K,p,L)=IncProj(K,o,R)
proof
  assume that
A1: not p on K and
A2: not p on L and
A3: not q on L and
A4: not q on R and
A5: c on K and
A6: c on L and
A7: c on R and
A8: K <> R;
  defpred P[object] means ex k being POINT of IPP st $1 = k & k on K;
  consider X such that
A9: for x being object holds x in X iff x in (the Points
  of IPP) qua set & P[x] from XBOOLE_0:sch 1;
A10: dom (IncProj(L,q,R)*IncProj(K,p,L)) c= (the Points of IPP) & rng (
  IncProj(L,q,R)*IncProj(K,p,L)) c= (the Points of IPP)
  proof
    set f = IncProj(L,q,R)*IncProj(K,p,L);
    dom f = dom IncProj (K,p,L) by A1,A2,A3,A4,Th22;
    hence dom (IncProj(L,q,R)*IncProj(K,p,L)) c= (the Points of IPP);
    rng f = rng IncProj(L,q,R) by A1,A2,A3,A4,Th22;
    hence thesis by RELAT_1:def 19;
  end;
A11: now
A12: now
A13:  X c= dom (IncProj (L,q,R)*IncProj(K,p,L))
      proof
        let e be object;
        assume
A14:    e in X;
        then reconsider e as Element of the Points of IPP by A9;
A15:    ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A14;
        dom (IncProj(L,q,R)*IncProj(K,p,L))=dom IncProj(K,p,L) by A1,A2,A3,A4
,Th22;
        hence thesis by A1,A2,A15,Def1;
      end;
      assume
A16:  L=R;
A17:  X c= dom IncProj (K,p,R)
      proof
        let e be object;
        assume
A18:    e in X;
        then reconsider e as POINT of IPP by A9;
        ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A18;
        hence thesis by A1,A2,A16,Def1;
      end;
A19:  for x being POINT of IPP st x in X holds (IncProj(L,q,R)*IncProj(K
      ,p,L)).x = IncProj (K,p,R).x
      proof
        let x be Element of the Points of IPP;
        assume
A20:    x in X;
        then
A21:    (IncProj(R,q,R)*IncProj(K,p,R)).x = IncProj(R,q,R).(IncProj (K,p
        ,R).x) by A16,A13,FUNCT_1:12;
A22:    x on K by A1,A2,A16,A17,A20,Def1;
        then reconsider y =IncProj(K,p,R).x as POINT of IPP by A1,A2,A16,Th19;
        y on R by A1,A2,A16,A22,Th20;
        hence thesis by A3,A16,A21,Th18;
      end;
      dom (IncProj(L,q,R)*IncProj (K,p,L)) c= X
      proof
        let e be object;
        assume e in dom (IncProj(L,q,R)*IncProj(K,p,L));
        then
A23:    e in dom IncProj(K,p,L) by A1,A2,A3,A4,Th22;
        then reconsider e as POINT of IPP;
        e on K by A1,A2,A23,Def1;
        hence thesis by A9;
      end;
      then
A24:  X = dom (IncProj(L,q,R)*IncProj(K,p,L)) by A13,XBOOLE_0:def 10;
A25:  (IncProj(L,q,R)*IncProj(K,p,L)) is PartFunc of the Points of IPP,
      the Points of IPP by A10,RELSET_1:4;
      take p;
      dom IncProj(K,p,R) c= X
      proof
        let e be object;
        assume
A26:    e in dom IncProj (K,p,R);
        then reconsider e as POINT of IPP;
        e on K by A1,A2,A16,A26,Def1;
        hence thesis by A9;
      end;
      then X = dom IncProj(K,p,R) by A17,XBOOLE_0:def 10;
      hence thesis by A1,A2,A16,A24,A19,A25,PARTFUN1:5;
    end;
    consider A such that
A27: p on A and
A28: q on A by INCPROJ:def 5;
    consider c1 being POINT of IPP such that
A29: c1 on K and
A30: c1 on A by INCPROJ:def 9;
    reconsider c2=IncProj(K,p,L).c1 as Element of the Points of IPP by A1,A2
,A29,Th19;
A31: c2 on L by A1,A2,A29,Th20;
    then reconsider c3=IncProj(L,q,R).c2 as POINT of IPP by A3,A4,Th19;
A32: c3 on R by A3,A4,A31,Th20;
    consider a1 being POINT of IPP such that
A33: a1 on K and
A34: c1<>a1 and
A35: c <>a1 by Th8;
    reconsider a2 =IncProj(K,p,L).a1 as POINT of IPP by A1,A2,A33,Th19;
A36: a2 on L by A1,A2,A33,Th20;
    then reconsider a3=IncProj(L,q,R).a2 as POINT of IPP by A3,A4,Th19;
A37: a3 on R by A3,A4,A36,Th20;
A38: not a3 on K
    proof
      assume a3 on K;
      then
A39:  c =a3 by A5,A7,A8,A37,INCPROJ:def 4;
      ex C st q on C & c on C by INCPROJ:def 5;
      then IncProj(L,q,R).c =a3 by A3,A4,A6,A7,A39,Def1;
      then
A40:  c =a2 by A3,A4,A6,A36,Th23;
      ex D st p on D & c on D by INCPROJ:def 5;
      then IncProj(K,p,L).c =a2 by A1,A2,A5,A6,A40,Def1;
      hence contradiction by A1,A2,A5,A33,A35,Th23;
    end;
    consider B such that
A41: a1 on B & a3 on B by INCPROJ:def 5;
    consider o being POINT of IPP such that
A42: o on A and
A43: o on B by INCPROJ:def 9;
A44: not a1 on R by A5,A7,A8,A33,A35,INCPROJ:def 4;
A45: not o on K & not o on R
    proof
A46:  now
        assume
A47:    o on R;
        then
A48:    o=a3 by A37,A41,A43,A44,INCPROJ:def 4;
        consider A2 being LINE of IPP such that
A49:    q on A2 and
A50:    c2 on A2 and
A51:    c3 on A2 by A3,A4,A31,A32,Def1;
        ex A1 being LINE of IPP st p on A1 & c1 on A1 & c2 on A1 by A1,A2,A29
,A31,Def1;
        then c2 on A by A1,A27,A29,A30,INCPROJ:def 4;
        then A=A2 by A3,A28,A31,A49,A50,INCPROJ:def 4;
        then o=c3 by A4,A42,A32,A47,A49,A51,INCPROJ:def 4;
        then c2=a2 by A3,A4,A36,A31,A48,Th23;
        hence contradiction by A1,A2,A29,A33,A34,Th23;
      end;
A52:  now
        assume
A53:    o on K;
        then o=c1 by A1,A27,A29,A30,A42,INCPROJ:def 4;
        hence contradiction by A33,A34,A41,A43,A38,A53,INCPROJ:def 4;
      end;
      assume not thesis;
      hence thesis by A52,A46;
    end;
    assume
A54: p<>q;
A55: now
      assume that
A56:  L<>R and
      K<>L;
A57:  X c= dom (IncProj (L,q,R)*IncProj(K,p,L))
      proof
        let e be object;
        assume
A58:    e in X;
        then reconsider e as POINT of IPP by A9;
A59:    ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A58;
        dom (IncProj(L,q,R)*IncProj(K,p,L)) = dom IncProj(K,p,L) by A1,A2,A3,A4
,Th22;
        hence thesis by A1,A2,A59,Def1;
      end;
A60:  for x being POINT of IPP st x in X holds (IncProj(L,q,R)*IncProj(K
      ,p,L)).x = IncProj (K,o,R).x
      proof
        let x be Element of the Points of IPP;
        assume
A61:    x in X;
A62:    now
          assume
A63:      x=c;
          then (IncProj(L,q,R)*IncProj(K,p,L)).c = IncProj(L,q,R).(IncProj(K,
          p,L).c) by A57,A61,FUNCT_1:12;
          then (IncProj(L,q,R)*IncProj(K,p,L)).c = IncProj(L,q,R).c by A1,A2,A5
,A6,Th24;
          then (IncProj(L,q,R)*IncProj(K,p,L)).c =c by A3,A4,A6,A7,Th24;
          hence thesis by A5,A7,A45,A63,Th24;
        end;
A64:    now
          assume that
          x<>c1 and
A65:      x<>c and
          x<>a1;
          (IncProj(L,q,R)*IncProj(K,p,L)).x = IncProj(K,o,R).x
          proof
A66:        a2<>a3
            proof
              assume a2=a3;
              then
A67:          IncProj(K,p,L).a1 = c by A6,A7,A36,A37,A56,INCPROJ:def 4;
              IncProj(K, p,L).c = c by A1,A2,A5,A6,Th24;
              hence contradiction by A1,A2,A5,A33,A35,A67,Th23;
            end;
A68:        (IncProj(L,q,R)*IncProj(K,p,L)).x = IncProj(L,q,R).(IncProj
            (K,p,L).x) by A57,A61,FUNCT_1:12;
A69:        a2<>c
            proof
              assume
A70:          a2=c;
              IncProj(K,p,L).c = c by A1,A2,A5,A6,Th24;
              hence contradiction by A1,A2,A5,A33,A35,A70,Th23;
            end;
A71:        a3 on R by A3,A4,A36,Th20;
A72:        dom ( IncProj (L,q,R)*IncProj(K,p,L))= dom IncProj(K,p,L) by A1,A2
,A3,A4,Th22;
            then
A73:        x on K by A1,A2,A57,A61,Def1;
            then reconsider y = IncProj(K,p,L).x as POINT of IPP by A1,A2,Th19;
A74:        y on L by A1,A2,A73,Th20;
            then reconsider z = IncProj(L,q,R).y as POINT of IPP by A3,A4,Th19;
            consider B3 being LINE of IPP such that
A75:        p on B3 & x on B3 & y on B3 by A1,A2,A73,A74,Def1;
            x on K by A1,A2,A57,A61,A72,Def1;
            then
A76:        c <>y by A1,A5,A65,A75,INCPROJ:def 4;
            consider A1 being LINE of IPP such that
A77:        q on A1 and
A78:        a2 on A1 & a3 on A1 by A3,A4,A36,A37,Def1;
A79:        {a2,a3,q} on A1 by A77,A78,INCSP_1:2;
A80:        z on R by A3,A4,A74,Th20;
            then consider B1 being LINE of IPP such that
A81:        q on B1 & y on B1 & z on B1 by A3,A4,A74,Def1;
            x on K by A1,A2,A57,A61,A72,Def1;
            then
A82:        {x,c,a1} on K by A5,A33,INCSP_1:2;
            consider A3 being Element of the Lines of IPP such that
A83:        p on A3 and
A84:        a1 on A3 and
A85:        a2 on A3 by A1,A2,A33,A36,Def1;
A86:        {a2,p,a1} on A3 by A83,A84,A85,INCSP_1:2;
            A1<>A3
            proof
              assume A1=A3;
              then A=A3 by A54,A27,A28,A77,A83,INCPROJ:def 4;
              hence contradiction by A1,A29,A30,A33,A34,A83,A84,INCPROJ:def 4;
            end;
            then
A87:        A1,L,A3 are_mutually_distinct by A2,A3,A77,A83,ZFMISC_1:def 5;
A88:        {a2,y,c} on L by A6,A36,A74,INCSP_1:2;
A89:        {p,y,x} on B3 by A75,INCSP_1:2;
            z on R by A3,A4,A74,Th20;
            then
A90:        {a3,z,c} on R by A7,A71,INCSP_1:2;
A91:        {p,o,q} on A & {a3,o,a1} on B by A27,A28,A41,A42,A43,INCSP_1:2;
A92:        a2<>p by A1,A2,A33,Th20;
            {y,z,q} on B1 by A81,INCSP_1:2;
            then consider O being LINE of IPP such that
A93:        {o,z,x} on O by A69,A66,A76,A88,A79,A86,A89,A91,A82,A90,A87,A92
,Th12;
A94:        o on O by A93,INCSP_1:2;
            x on O & z on O by A93,INCSP_1:2;
            hence thesis by A45,A73,A80,A94,A68,Def1;
          end;
          hence thesis;
        end;
A95:    now
          assume
A96:      x=c1;
          (IncProj(L,q,R)*IncProj(K,p,L)).x = IncProj (K,o,R).x
          proof
A97:        (IncProj(L,q,R)*IncProj(K,p,L)).c1 =c3 by A57,A61,A96,FUNCT_1:12;
            consider A2 being LINE of IPP such that
A98:        q on A2 & c2 on A2 and
A99:        c3 on A2 by A3,A4,A31,A32,Def1;
            ex A1 being Element of the Lines of IPP st p on A1 & c1 on
            A1 & c2 on A1 by A1,A2,A29,A31,Def1;
            then c2 on A by A1,A27,A29,A30,INCPROJ:def 4;
            then A=A2 by A3,A28,A31,A98,INCPROJ:def 4;
            hence thesis by A29,A30,A42,A32,A45,A96,A97,A99,Def1;
          end;
          hence thesis;
        end;
        now
          assume
A100:     x=a1;
          then (IncProj(L,q,R)*IncProj(K,p,L)).a1 =a3 by A57,A61,FUNCT_1:12;
          hence thesis by A33,A37,A41,A43,A45,A100,Def1;
        end;
        hence thesis by A62,A95,A64;
      end;
A101: dom IncProj(K,o,R) c= X
      proof
        let e be object;
        assume
A102:   e in dom IncProj (K,o,R);
        then reconsider e as POINT of IPP;
        e on K by A45,A102,Def1;
        hence thesis by A9;
      end;
      X c= dom IncProj(K,o,R)
      proof
        let e be object;
        assume
A103:   e in X;
        then reconsider e as POINT of IPP by A9;
        ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A103;
        hence thesis by A45,Def1;
      end;
      then
A104: X = dom IncProj(K,o,R ) by A101,XBOOLE_0:def 10;
A105: (IncProj(L,q,R)* IncProj(K,p,L)) is PartFunc of the Points of IPP,
      the Points of IPP by A10,RELSET_1:4;
      dom (IncProj(L,q,R)*IncProj(K,p,L)) c= X
      proof
        let e be object;
        assume e in dom (IncProj(L,q,R)*IncProj(K,p,L));
        then
A106:   e in dom IncProj (K,p,L) by A1,A2,A3,A4,Th22;
        then reconsider e as POINT of IPP;
        e on K by A1,A2,A106,Def1;
        hence thesis by A9;
      end;
      then X = dom (IncProj(L,q,R)*IncProj(K,p,L)) by A57,XBOOLE_0:def 10;
      hence thesis by A45,A60,A104,A105,PARTFUN1:5;
    end;
    now
A107: X c= dom (IncProj (L,q,R)*IncProj(K,p,L))
      proof
        let e be object;
        assume
A108:   e in X;
        then reconsider e as Element of the Points of IPP by A9;
A109:   ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A108;
        dom (IncProj(L,q,R)*IncProj(K,p,L)) = dom IncProj(K,p,L) by A1,A2,A3,A4
,Th22;
        hence thesis by A1,A2,A109,Def1;
      end;
      dom (IncProj(L,q,R)*IncProj(K,p,L)) c= X
      proof
        let e be object;
        assume e in dom (IncProj(L,q,R)*IncProj(K,p,L));
        then
A110:   e in dom IncProj (K,p,L) by A1,A2,A3,A4,Th22;
        then reconsider e as POINT of IPP;
        e on K by A1,A2,A110,Def1;
        hence thesis by A9;
      end;
      then
A111: X = dom (IncProj(L,q,R)*IncProj(K,p,L)) by A107,XBOOLE_0:def 10;
A112: (IncProj(L,q,R)*IncProj(K,p,L)) is PartFunc of the Points of IPP,
      the Points of IPP by A10,RELSET_1:4;
      assume
A113: K=L;
A114: X c= dom IncProj(K,q,R)
      proof
        let e be object;
        assume
A115:   e in X;
        then reconsider e as POINT of IPP by A9;
        ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A115;
        hence thesis by A3,A4,A113,Def1;
      end;
A116: for x being POINT of IPP st x in X holds (IncProj(L,q,R)*IncProj(K
      ,p,L)).x = IncProj (K,q,R).x
      proof
        let x be Element of the Points of IPP;
        assume x in X;
        then x on K & (IncProj(K,q,R)*IncProj(K,p,K)).x = IncProj(K,q,R).(
        IncProj (K,p,K) .x) by A3,A4,A113,A107,A114,Def1,FUNCT_1:12;
        hence thesis by A1,A113,Th18;
      end;
      take q;
      dom IncProj(K,q,R) c= X
      proof
        let e be object;
        assume
A117:   e in dom IncProj (K,q,R);
        then reconsider e as POINT of IPP;
        e on K by A3,A4,A113,A117,Def1;
        hence thesis by A9;
      end;
      then X = dom IncProj(K,q,R) by A114,XBOOLE_0:def 10;
      hence thesis by A3,A4,A113,A111,A116,A112,PARTFUN1:5;
    end;
    hence thesis by A12,A55;
  end;
  now
A118: X c= dom (IncProj(L,q,R)*IncProj (K,p,L))
    proof
      let e be object;
      assume
A119: e in X;
      then reconsider e as POINT of IPP by A9;
A120: ex e9 being Element of the Points of IPP st e=e9 & e9 on K by A9,A119;
      dom (IncProj(L,q,R)*IncProj(K,p,L))=dom IncProj(K,p,L) by A1,A2,A3,A4
,Th22;
      hence thesis by A1,A2,A120,Def1;
    end;
    assume
A121: p=q;
A122: X c= dom IncProj (K,p,R)
    proof
      let e be object;
      assume
A123: e in X;
      then reconsider e as POINT of IPP by A9;
      ex e9 being POINT of IPP st e=e9 & e9 on K by A9,A123;
      hence thesis by A1,A4,A121,Def1;
    end;
A124: for x being POINT of IPP st x in X holds (IncProj(L,q,R)*IncProj(K,p,
    L)).x = IncProj(K,p,R).x
    proof
      let x be POINT of IPP;
      assume
A125: x in X;
      then
A126: (IncProj(L,p,R)*IncProj(K,p,L)).x = IncProj(L,p,R).(IncProj(K,p,L).
      x) by A121,A118,FUNCT_1:12;
A127: x on K by A1,A4,A121,A122,A125,Def1;
      then reconsider y = IncProj(K,p,L).x as POINT of IPP by A1,A2,Th19;
A128: y on L by A1,A2,A127,Th20;
      then reconsider z = IncProj (L,p,R).y as POINT of IPP by A2,A4,A121,Th19;
      consider A such that
A129: p on A & y on A by INCPROJ:def 5;
A130: z on R by A2,A4,A121,A128,Th20;
      then consider C such that
A131: p on C & y on C and
A132: z on C by A2,A4,A121,A128,Def1;
A133: A=C by A2,A128,A129,A131,INCPROJ:def 4;
      consider B such that
A134: p on B and
A135: x on B and
A136: y on B by A1,A2,A127,A128,Def1;
      A=B by A2,A128,A129,A134,A136,INCPROJ:def 4;
      hence thesis by A1,A4,A121,A127,A126,A134,A135,A130,A132,A133,Def1;
    end;
    dom (IncProj(L,q,R)*IncProj(K,p,L)) c= X
    proof
      let e be object;
      assume e in dom (IncProj(L,q,R)*IncProj(K,p,L));
      then
A137: e in dom IncProj (K,p,L) by A1,A2,A3,A4,Th22;
      then reconsider e as POINT of IPP;
      e on K by A1,A2,A137,Def1;
      hence thesis by A9;
    end;
    then
A138: X = dom (IncProj(L,q,R)*IncProj(K,p,L)) by A118,XBOOLE_0:def 10;
A139: (IncProj(L,q,R)*IncProj(K,p,L))is PartFunc of the Points of IPP,the
    Points of IPP by A10,RELSET_1:4;
    dom IncProj(K,p,R) c= X
    proof
      let e be object;
      assume
A140: e in dom IncProj (K,p,R);
      then reconsider e as POINT of IPP;
      e on K by A1,A4,A121,A140,Def1;
      hence thesis by A9;
    end;
    then X = dom IncProj(K,p,R) by A122,XBOOLE_0:def 10;
    hence thesis by A1,A4,A121,A138,A124,A139,PARTFUN1:5;
  end;
  hence thesis by A11;
end;
