reserve n for Nat;
reserve K for Field;
reserve a,b,c,d,e,f,g,h,i,a1,b1,c1,d1,e1,f1,g1,h1,i1 for Element of K;
reserve M,N for Matrix of 3,K;
reserve p for FinSequence of REAL;
reserve a,b,c,d,e,f for Real;
reserve u,u1,u2 for non zero Element of TOP-REAL 3;
reserve P for Element of ProjectiveSpace TOP-REAL 3;
reserve a,b,c,d,e,f,g,h,i for Element of F_Real;
reserve M for Matrix of 3,F_Real;
reserve e1,e2,e3,f1,f2,f3 for Element of F_Real;
reserve MABC,MAEF,MDBF,MDEC,MDEF,MDBC,MAEC,MABF,
        MABE,MACF,MBDF,MCDE,MACE,MBDE,MCDF for Matrix of 3,F_Real;
reserve r1,r2 for Real;
reserve p1,p2,p3,p4,p5,p6 for Point of TOP-REAL 3;
reserve p7,p8,p9 for Point of TOP-REAL 3;

theorem Th31:
  for PCPP being CollProjectiveSpace
  for c1,c2,c3,c4,c5,c6,c7,c8,c9 being Element of PCPP st
  not c1,c2,c4 are_collinear & not c1,c2,c5 are_collinear &
  not c1,c6,c4 are_collinear & not c1,c6,c5 are_collinear &
  not c2,c6,c4 are_collinear & not c3,c4,c2 are_collinear &
  not c3,c4,c6 are_collinear & not c3,c5,c2 are_collinear &
  not c3,c5,c6 are_collinear & not c4,c5,c2 are_collinear &
  c1,c4,c7 are_collinear & c1,c5,c8 are_collinear &
  c2,c3,c7 are_collinear & c2,c5,c9 are_collinear &
  c6,c3,c8 are_collinear & c6,c4,c9 are_collinear holds
  not c9,c2,c4 are_collinear & not c1,c4,c9 are_collinear &
  not c2,c3,c9 are_collinear & not c2,c4,c7 are_collinear &
  not c2,c5,c8 are_collinear & not c2,c9,c8 are_collinear &
  not c2,c9,c7 are_collinear & not c6,c4,c8 are_collinear &
  not c6,c5,c8 are_collinear & not c4,c9,c8 are_collinear &
  not c4,c9,c7 are_collinear
  proof
    let PCPP be CollProjectiveSpace;
    let c1,c2,c3,c4,c5,c6,c7,c8,c9 being Element of PCPP;
    assume that
A11: not c1,c2,c4 are_collinear and
A12: not c1,c2,c5 are_collinear and
A13: not c1,c6,c4 are_collinear and
A14: not c1,c6,c5 are_collinear and
A15: not c2,c6,c4 are_collinear and
A16: not c3,c4,c2 are_collinear and
A17: not c3,c4,c6 are_collinear and
A18: not c3,c5,c2 are_collinear and
A19: not c3,c5,c6 are_collinear and
A20: not c4,c5,c2 are_collinear and
A21: c1,c4,c7 are_collinear and
A22: c1,c5,c8 are_collinear and
A23: c2,c3,c7 are_collinear and
A24: c2,c5,c9 are_collinear and
A25: c6,c3,c8 are_collinear and
A26: c6,c4,c9 are_collinear and
A27: c9,c2,c4 are_collinear or c1,c4,c9 are_collinear or
    c2,c3,c9 are_collinear or c2,c4,c7 are_collinear or c2,c5,c8
    are_collinear or c2,c9,c8 are_collinear or c2,c9,c7 are_collinear or
    c6,c4,c8 are_collinear or c6,c5,c8 are_collinear or c4,c9,c8
    are_collinear or c4,c9,c7 are_collinear;
A34: for v102,v103,v100,v104 being Element of PCPP holds v100=v104
      or not v104,v100,v102
    are_collinear or  not v104,v100,v103 are_collinear or v102,v103,v104
    are_collinear
    proof let v102,v103,v100,v104 being Element of PCPP;
      v104,v100,v104 are_collinear by COLLSP:5;
      hence thesis by COLLSP:3;
    end;
A38: for v102,v104,v100,v103 being Element of PCPP holds v100=v103 or
      not v103,v100,v102
    are_collinear or  not v103,v100,v104 are_collinear or v102,v103,v104
    are_collinear
    proof let v102,v104,v100,v103 being Element of PCPP;
      v103,v100,v103 are_collinear by COLLSP:5;
      hence thesis by COLLSP:3;
    end;
A47: not c5=c2 by COLLSP:2,A12;
A49: not c6,c4,c1 are_collinear by HESSENBE:1,A13;
A51: not c4,c1,c6 are_collinear by HESSENBE:1,A13;
A54: not c6=c4 by A13,COLLSP:2;
A61: not c6,c4,c2 are_collinear by HESSENBE:1,A15;
A67: not c6,c3,c4 are_collinear by HESSENBE:1,A17;
A69: not c2,c3,c5 are_collinear by HESSENBE:1,A18;
A75: c4,c7,c1 are_collinear by A21,HESSENBE:1;
A79: c2,c7,c3 are_collinear by A23,COLLSP:4;
A81: c9,c2,c5 are_collinear by A24,HESSENBE:1;
A83: c2,c9,c5 are_collinear by A24,COLLSP:4;
A85: c6,c8,c3 are_collinear by A25,COLLSP:4;
A89: c4,c9,c6 are_collinear by A26,HESSENBE:1;
A98: c1,c4,c9 are_collinear or c2,c3,c9 are_collinear or
      c2,c4,c7 are_collinear or c2,c5,c8 are_collinear or
      c2,c9,c8 are_collinear or c2,c9,c7 are_collinear or
      c6,c4,c8 are_collinear or c6,c5,c8 are_collinear or
      c4,c9,c8 are_collinear or c4,c9,c7 are_collinear or
      c9,c4,c2 are_collinear by A27,COLLSP:4;
A101: for v102,v103,v104,v101 being Element of PCPP holds v104=v101 or
      not v101,v104,v102 are_collinear or
      not v101,v104,v103 are_collinear or v102,v103,v104 are_collinear
      proof
        let v102,v103,v104,v101 being Element of PCPP;
        v101,v104,v104 are_collinear by COLLSP:2;
        hence thesis by COLLSP:3;
      end;
A105: for v103,v104,v102,v101 being Element of PCPP holds v102=v101 or
      not v101,v102,v103 are_collinear or  not v101,v102,v104 are_collinear
      or v102,v103,v104 are_collinear
      proof
        let v103,v104,v102,v101 being Element of PCPP;
        v101,v102,v102 are_collinear by COLLSP:2;
        hence thesis by COLLSP:3;
      end;
A121: c7,c4,c1 are_collinear by A21,HESSENBE:1;
A124: for v2,v101,v100 being Element of PCPP holds v101=v100 or
      not v100,v101,v2 are_collinear or v2,v101,v100 are_collinear
      proof
        let v2,v101,v100 being Element of PCPP;
        v101=v100 or not v100,v101,v2 are_collinear or
          not v100,v101,v101 are_collinear or v2,v101,v100 are_collinear
          by A34;
        hence thesis by COLLSP:2;
      end;
A130: c2,c5,c5 are_collinear by COLLSP:2;
A133: not c2,c5,c4 are_collinear by A130,A47,A34,A20;
A144: c8,c5,c1 are_collinear by A22,HESSENBE:1;
A147: for v0 being Element of PCPP holds c7=c4 or
        not c4,c7,v0 are_collinear or v0,c4,c1 are_collinear by A75,A38;
A158: for v0 being Element of PCPP holds c7=c2 or
        not c2,c7,v0 are_collinear or c2,c3,v0 are_collinear by A79,HESSENBE:2;
A161: for v0 being Element of PCPP holds c9=c2 or
        not c2,c9,v0 are_collinear or c2,v0,c5 are_collinear by A83,HESSENBE:2;
A169: c9,c4,c6 are_collinear by A26,HESSENBE:1;
A175: for v0 being Element of PCPP holds c9=c4 or
        not c4,c9,v0 are_collinear or c4,v0,c6 are_collinear by A89,HESSENBE:2;
A203: c5=c6 or  not c6,c5,c8 are_collinear or c8,c5,c6 are_collinear by A124;
A213: c9=c4 or  not c4,c9,c8 are_collinear or c6,c8,c4 are_collinear
        by A89,A34;
A234: c5=c2 or  not c2,c5,c8 are_collinear or c8,c5,c2 are_collinear by A124;
A241: c4=c2 or  not c2,c4,c7 are_collinear or c7,c4,c2 are_collinear by A124;
A247: c9=c4 or c1,c4,c9 are_collinear or c2,c3,c9 are_collinear or
        c8=c5 or c8=c6 or c7=c4 or c9=c2 or c7=c2
        by A11,A121,A105,COLLSP:2,A12,A69,A158,A49,A175,A67,
           A54,A26,A14,A144,A15,A98,A169,A101,A203,A213,A85,
           A147,A83,HESSENBE:2,A161,A241,A234;
A248: not c2,c3,c9 are_collinear or c9,c2,c3 are_collinear by HESSENBE:1;
     c4=c1 or  not c1,c4,c9 are_collinear or c9,c4,c1 are_collinear by A124;
     hence contradiction
        by A11,A21,A16,HESSENBE:1,A23,A19,A25,A14,A22,A61,A26,
           A133,A24,A51,A169,COLLSP:2,A69,A247,A248,A81,A105;
   end;
