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;

theorem Th12:
  for IPP being Desarguesian IncProjSp holds for o,b1,a1,b2,a2,b3,
a3,r,s,t being POINT of IPP, C1,C2,C3,A1,A2,A3,B1,B2,B3 being LINE of IPP st {o
,b1,a1} on C1 & {o,a2,b2} on C2 & {o,a3,b3} on C3 & {a3,a2,t} on A1 & {a3,r,a1}
on A2 & {a2,s,a1} on A3 & {t,b2,b3} on B1 & {b1,r,b3} on B2 & {b1,s,b2} on B3 &
  C1,C2,C3 are_mutually_distinct & o<>a3 & o<>b1 & o<>b2 & a2<>b2 ex O being
  LINE of IPP st {r,s,t} on O
proof
  let IPP be Desarguesian IncProjSp;
  let o,b1,a1,b2,a2,b3,a3,r,s,t be POINT of IPP, C1,C2,C3,A1,A2,A3,B1,B2,B3 be
  LINE of IPP;
  assume that
A1: {o,b1,a1} on C1 and
A2: {o,a2,b2} on C2 and
A3: {o,a3,b3} on C3 and
A4: {a3,a2,t} on A1 and
A5: {a3,r,a1} on A2 and
A6: {a2,s,a1} on A3 and
A7: {t,b2,b3} on B1 and
A8: {b1,r,b3} on B2 and
A9: {b1,s,b2} on B3 and
A10: C1,C2,C3 are_mutually_distinct and
A11: o<>a3 and
A12: o<>b1 and
A13: o<>b2 and
A14: a2<>b2;
A15: r on A2 & r on B2 by A5,A8,INCSP_1:2;
A16: s on B3 by A9,INCSP_1:2;
A17: a3 on A1 by A4,INCSP_1:2;
A18: b3 on B1 by A7,INCSP_1:2;
A19: b2 on B1 by A7,INCSP_1:2;
A20: b3 on C3 by A3,INCSP_1:2;
A21: a1 on C1 by A1,INCSP_1:2;
A22: o on C2 by A2,INCSP_1:2;
A23: s on A3 by A6,INCSP_1:2;
A24: a1 on A3 by A6,INCSP_1:2;
A25: a1 on A2 by A5,INCSP_1:2;
A26: t on A1 by A4,INCSP_1:2;
A27: b1 on B2 by A8,INCSP_1:2;
A28: t on B1 by A7,INCSP_1:2;
A29: b1 on B3 by A9,INCSP_1:2;
A30: now
    assume that
    o<>b3 and
    o<>a1 and
    o<>a2 and
A31: a1=b1;
A32: now
A33:  b3 on B2 by A8,INCSP_1:2;
      consider O being LINE of IPP such that
A34:  t on O & s on O by INCPROJ:def 5;
      assume
A35:  a3<>b3;
      take O;
A36:  b2 on C2 & o on C2 by A2,INCSP_1:2;
A37:  o on C1 & a2 on C2 by A1,A2,INCSP_1:2;
A38:  b1 on B3 & b2 on B3 by A9,INCSP_1:2;
A39:  b1 on A2 & a3 on A2 by A5,A31,INCSP_1:2;
A40:  a3 on C3 & b3 on C3 by A3,INCSP_1:2;
A41:  o on C1 & o on C3 by A1,A3,INCSP_1:2;
A42:  a2 on A3 by A6,INCSP_1:2;
      C1<>C2 & b1 on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
      then A3<>B3 by A12,A14,A37,A36,A38,A42,Th10;
      then
A43:  b1=s by A23,A24,A29,A16,A31,INCPROJ:def 4;
      C1<>C3 & b1 on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
      then A2<>B2 by A12,A35,A41,A40,A39,A33,Th10;
      then s = r by A25,A27,A15,A31,A43,INCPROJ:def 4;
      hence {r,s,t} on O by A34,INCSP_1:2;
    end;
    now
A44:  a2 on A3 & b1 on A3 by A6,A31,INCSP_1:2;
A45:  C2<>C3 & o on C2 by A2,A10,INCSP_1:2,ZFMISC_1:def 5;
A46:  a2 on C2 & b2 on C2 by A2,INCSP_1:2;
A47:  o on C1 & a2 on C2 by A1,A2,INCSP_1:2;
A48:  b2 on B3 by A9,INCSP_1:2;
A49:  a2 on A1 & b2 on B1 by A4,A7,INCSP_1:2;
A50:  o on C3 & b3 on C3 by A3,INCSP_1:2;
      assume
A51:  a3=b3;
      then b3 on A1 by A4,INCSP_1:2;
      then A1<>B1 by A11,A14,A51,A45,A46,A50,A49,Th10;
      then
A52:  t=b3 by A17,A26,A28,A18,A51,INCPROJ:def 4;
      take B2;
A53:  b2 on C2 & o on C2 by A2,INCSP_1:2;
      C1<>C2 & b1 on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
      then A3<>B3 by A12,A14,A47,A53,A44,A48,Th10;
      then {s,r,t} on B2 by A8,A23,A24,A29,A16,A31,A52,INCPROJ:def 4;
      hence {r,s,t} on B2 by Th11;
    end;
    hence thesis by A32;
  end;
A54: o on C1 by A1,INCSP_1:2;
A55: C1<>C2 by A10,ZFMISC_1:def 5;
A56: b1 on C1 by A1,INCSP_1:2;
A57: a2 on A3 by A6,INCSP_1:2;
A58: b2 on C2 by A2,INCSP_1:2;
A59: C2<>C3 by A10,ZFMISC_1:def 5;
A60: a3 on C3 by A3,INCSP_1:2;
A61: b3 on B2 by A8,INCSP_1:2;
A62: a3 on A2 by A5,INCSP_1:2;
A63: a2 on A1 by A4,INCSP_1:2;
A64: o on C3 by A3,INCSP_1:2;
A65: b2 on B3 by A9,INCSP_1:2;
A66: now
    assume that
A67: o<>b3 and
A68: o<>a1 and
A69: o=a2;
A70: now
      assume
A71:  a1=b1;
A72:  now
A73:    a2 on A1 & b2 on B1 by A4,A7,INCSP_1:2;
A74:    b2 on C2 & b3 on C3 by A2,A3,INCSP_1:2;
A75:    b2 on C2 & a2 on A3 by A2,A6,INCSP_1:2;
A76:    o on C2 & b1 on C1 by A1,A2,INCSP_1:2;
A77:    C1<>C2 & o on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
        assume
A78:    a3=b3;
        take B2;
A79:    b2 on B3 by A9,INCSP_1:2;
        b1 on A3 & a2 on C2 by A2,A6,A71,INCSP_1:2;
        then A3<>B3 by A12,A14,A77,A76,A75,A79,Th10;
        then
A80:    s=b1 by A23,A24,A29,A16,A71,INCPROJ:def 4;
A81:    o on C3 & a2 on C2 by A2,A3,INCSP_1:2;
A82:    C2<>C3 & o on C2 by A2,A10,INCSP_1:2,ZFMISC_1:def 5;
        b3 on A1 by A4,A78,INCSP_1:2;
        then A1<>B1 by A11,A14,A78,A82,A81,A74,A73,Th10;
        then {s,r,t} on B2 by A8,A17,A26,A28,A18,A78,A80,INCPROJ:def 4;
        hence {r,s,t} on B2 by Th11;
      end;
      now
A83:    o on C3 & a1 on C1 by A1,A3,INCSP_1:2;
A84:    a1 on A2 & a3 on A2 by A5,INCSP_1:2;
A85:    b3 on B2 by A8,INCSP_1:2;
        consider O being LINE of IPP such that
A86:    t on O & s on O by INCPROJ:def 5;
        assume
A87:    a3<>b3;
        take O;
A88:    a3 on C3 & b3 on C3 by A3,INCSP_1:2;
        C1<>C3 & o on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
        then A2<>B2 by A12,A71,A87,A83,A88,A84,A85,Th10;
        then
A89:    r=b1 by A25,A27,A15,A71,INCPROJ:def 4;
        A3=C1 & B3<>C1 by A13,A54,A21,A22,A58,A57,A24,A65,A55,A68,A69,
INCPROJ:def 4;
        then r=s by A23,A24,A29,A16,A71,A89,INCPROJ:def 4;
        hence {r,s,t} on O by A86,INCSP_1:2;
      end;
      hence thesis by A72;
    end;
    now
      assume
A90:  a1<>b1;
A91:  now
A92:    B1<>C3 by A13,A22,A58,A64,A19,A59,INCPROJ:def 4;
A93:    o on C3 & a1 on C1 by A1,A3,INCSP_1:2;
        consider O being Element of the Lines of IPP such that
A94:    t on O & s on O by INCPROJ:def 5;
A95:    C1<>C3 & o on C1 by A1,A10,INCSP_1:2,ZFMISC_1:def 5;
A96:    a1 on A2 & b1 on B2 by A5,A8,INCSP_1:2;
A97:    b1 on C1 & b3 on C3 by A1,A3,INCSP_1:2;
        assume
A98:    a3=b3;
        then b3 on A2 by A5,INCSP_1:2;
        then A2<>B2 by A11,A90,A98,A95,A93,A97,A96,Th10;
        then
A99:    r=b3 by A62,A15,A61,A98,INCPROJ:def 4;
        take O;
        A1=C3 by A64,A60,A17,A63,A67,A69,A98,INCPROJ:def 4;
        then r=t by A20,A26,A28,A18,A92,A99,INCPROJ:def 4;
        hence {r,s,t} on O by A94,INCSP_1:2;
      end;
      now
        assume a3<>b3;
        take B2;
        A3=C1 & B3<>C1 by A13,A54,A21,A22,A58,A57,A24,A65,A55,A68,A69,
INCPROJ:def 4;
        then
A100:   b1=s by A56,A23,A29,A16,INCPROJ:def 4;
        A1=C3 & B1<>C3 by A11,A13,A22,A58,A64,A60,A17,A63,A19,A59,A69,
INCPROJ:def 4;
        then {s,r,t} on B2 by A8,A20,A26,A28,A18,A100,INCPROJ:def 4;
        hence {r,s,t} on B2 by Th11;
      end;
      hence thesis by A91;
    end;
    hence thesis by A70;
  end;
A101: C3<>C1 by A10,ZFMISC_1:def 5;
A102: a2 on C2 by A2,INCSP_1:2;
A103: now
    assume that
A104: o<>b3 and
A105: o=a1;
A106: now
A107: now
A108:   B2<>C3 by A12,A54,A56,A64,A27,A101,INCPROJ:def 4;
        assume
A109:   a3=b3;
        then A2=C3 by A64,A60,A62,A25,A104,A105,INCPROJ:def 4;
        then
A110:   r=b3 by A20,A15,A61,A108,INCPROJ:def 4;
A111:   b2 on C2 & b3 on C3 by A2,A3,INCSP_1:2;
A112:   o on C3 & a2 on C2 by A2,A3,INCSP_1:2;
A113:   a2 on A1 & b2 on B1 by A4,A7,INCSP_1:2;
        consider O being LINE of IPP such that
A114:   t on O & s on O by INCPROJ:def 5;
A115:   C2<>C3 & o on C2 by A2,A10,INCSP_1:2,ZFMISC_1:def 5;
        take O;
        b3 on A1 by A4,A109,INCSP_1:2;
        then A1<>B1 by A11,A14,A109,A115,A112,A111,A113,Th10;
        then r=t by A17,A26,A28,A18,A109,A110,INCPROJ:def 4;
        hence {r,s,t} on O by A114,INCSP_1:2;
      end;
      assume
A116: o<>a2;
      now
        assume a3<>b3;
        take B1;
        A3=C2 & B3<>C2 by A12,A54,A56,A22,A102,A57,A24,A29,A55,A105,A116,
INCPROJ:def 4;
        then
A117:   s=b2 by A58,A23,A16,A65,INCPROJ:def 4;
        A2=C3 & B2<>C3 by A11,A12,A54,A56,A64,A60,A62,A25,A27,A101,A105,
INCPROJ:def 4;
        then {t,s,r} on B1 by A7,A20,A15,A61,A117,INCPROJ:def 4;
        hence {r,s,t} on B1 by Th11;
      end;
      hence thesis by A107;
    end;
    now
      assume o=a2;
      then
A118: A1=C3 by A11,A64,A60,A17,A63,INCPROJ:def 4;
      A2=C3 & B2<>C3 by A11,A12,A54,A56,A64,A60,A62,A25,A27,A101,A105,
INCPROJ:def 4;
      then
A119: r=b3 by A20,A15,A61,INCPROJ:def 4;
      consider O being LINE of IPP such that
A120: t on O & s on O by INCPROJ:def 5;
      take O;
      B1<>C3 by A13,A22,A58,A64,A19,A59,INCPROJ:def 4;
      then r=t by A20,A26,A28,A18,A118,A119,INCPROJ:def 4;
      hence {r,s,t} on O by A120,INCSP_1:2;
    end;
    hence thesis by A106;
  end;
A121: C1<>B3 by A13,A54,A22,A58,A65,A55,INCPROJ:def 4;
A122: now
    assume
A123: o=b3;
A124: now
      assume
A125: a1=o;
A126: now
        assume o=a2;
        then
A127:   A1=C3 by A11,A64,A60,A17,A63,INCPROJ:def 4;
        A2=C3 & B2=C1 by A11,A12,A54,A56,A64,A60,A62,A25,A27,A61,A123,A125,
INCPROJ:def 4;
        then
A128:   o=r by A54,A64,A15,A101,INCPROJ:def 4;
        consider O being LINE of IPP such that
A129:   t on O & s on O by INCPROJ:def 5;
        take O;
        B1=C2 by A13,A22,A58,A19,A18,A123,INCPROJ:def 4;
        then r=t by A22,A64,A26,A28,A59,A128,A127,INCPROJ:def 4;
        hence {r,s,t} on O by A129,INCSP_1:2;
      end;
      now
        B2=C1 & A2=C3 by A11,A12,A54,A56,A64,A60,A62,A25,A27,A61,A123,A125,
INCPROJ:def 4;
        then
A130:   r=o by A54,A64,A15,A101,INCPROJ:def 4;
        assume o<>a2;
        then
A131:   C2=A3 by A22,A102,A57,A24,A125,INCPROJ:def 4;
        take C2;
        C2=B1 by A13,A22,A58,A19,A18,A123,INCPROJ:def 4;
        hence {r,s,t} on C2 by A22,A23,A28,A131,A130,INCSP_1:2;
      end;
      hence thesis by A126;
    end;
    now
      assume
A132: a1<>o;
A133: now
        assume
A134:   o=a2;
        then
A135:   A1=C3 by A11,A64,A60,A17,A63,INCPROJ:def 4;
        take B2;
        C1=A3 by A54,A21,A57,A24,A132,A134,INCPROJ:def 4;
        then
A136:   b1=s by A56,A23,A29,A16,A121,INCPROJ:def 4;
        B1=C2 by A13,A22,A58,A19,A18,A123,INCPROJ:def 4;
        then {s,r,t} on B2 by A8,A20,A26,A28,A18,A59,A136,A135,INCPROJ:def 4;
        hence {r,s,t} on B2 by Th11;
      end;
      now
        assume o<>a2;
        take A3;
        B2=C1 & A2<>C1 by A11,A12,A54,A56,A64,A60,A62,A27,A61,A101,A123,
INCPROJ:def 4;
        then
A137:   r=a1 by A21,A25,A15,INCPROJ:def 4;
        B1=C2 & A1<>C2 by A11,A13,A22,A58,A64,A60,A17,A19,A18,A59,A123,
INCPROJ:def 4;
        then {t,s,r} on A3 by A6,A102,A63,A26,A28,A137,INCPROJ:def 4;
        hence {r,s,t} on A3 by Th11;
      end;
      hence thesis by A133;
    end;
    hence thesis by A124;
  end;
  now
A138: b2 on B1 & a2 on C2 by A2,A7,INCSP_1:2;
A139: o on C3 & b3 on C3 by A3,INCSP_1:2;
A140: a1 on A2 & b1 on B2 by A5,A8,INCSP_1:2;
A141: b2 on C2 & o on C2 by A2,INCSP_1:2;
A142: b3 on B1 & t on A1 by A4,A7,INCSP_1:2;
A143: t on B1 by A7,INCSP_1:2;
    consider O being Element of the Lines of IPP such that
A144: t on O & s on O by INCPROJ:def 5;
    assume that
A145: o<>b3 and
    o<>a1 and
    o<>a2 and
A146: a1<>b1 and
A147: a3=b3;
    take O;
A148: C1<>C3 by A10,ZFMISC_1:def 5;
    b3 on A2 by A5,A147,INCSP_1:2;
    then A2<>B2 by A54,A56,A21,A64,A20,A145,A146,A140,A148,Th10;
    then
A149: b3 = r by A62,A15,A61,A147,INCPROJ:def 4;
A150: b3 on A1 by A4,A147,INCSP_1:2;
    C2<>C3 & a2 on A1 by A4,A10,INCSP_1:2,ZFMISC_1:def 5;
    then A1<>B1 by A14,A145,A150,A138,A141,A139,Th10;
    then r = t by A150,A142,A143,A149,INCPROJ:def 4;
    hence {r,s,t} on O by A144,INCSP_1:2;
  end;
  hence
  thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A122,A103,A66,A30,
INCPROJ:def 13;
end;
