reserve a,b,c,d,e,f,g,h,i for Real,
                        M for Matrix of 3,REAL;
reserve                           PCPP for CollProjectiveSpace,
        c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 for Element of PCPP;
reserve o,p1,p2,p3,q1,q2,q3,r1,r2,r3 for Element of ProjectiveSpace TOP-REAL 3;
reserve v0,v1,v2,v3,v4,
        c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,
        v100,v101,v102,v103 for Element of ProjectiveSpace TOP-REAL 3;

theorem Th19:
  c1 <> c2 & c1 <> c3 & c2 <> c3 & c2 <> c4 & c3 <> c4 & c1 <> c5 & c1 <> c6 &
  c5 <> c6 & c5 <> c7 & c6 <> c7 & (not c1,c4,c7 are_collinear) &
  (c1,c4,c2 are_collinear) & (c1,c4,c3 are_collinear) &
  (c1,c7,c5 are_collinear) & (c1,c7,c6 are_collinear) &
  (c4,c5,c8 are_collinear) & (c7,c2,c8 are_collinear) &
  (c4,c6,c9 are_collinear) & (c3,c7,c9 are_collinear) &
  (c2,c6,c10 are_collinear) & (c3,c5,c10 are_collinear)
  implies not(
  (c4,c2,c7 are_collinear or c4,c3,c7 are_collinear or c2,c3,c7
  are_collinear or c4,c2,c5 are_collinear or c4,c2,c6 are_collinear or
  c4,c3,c5 are_collinear or c4,c3,c6 are_collinear or c2,c7,c5
  are_collinear or c2,c7,c6 are_collinear or c3,c7,c5 are_collinear or
  c3,c7,c6 are_collinear or c2,c3,c5 are_collinear or c2,c3,c6
  are_collinear or c7,c5,c4 are_collinear or c7,c6,c4 are_collinear or
  c5,c6,c4 are_collinear or c5,c6,c2 are_collinear or not c4,c5,c8
  are_collinear or not c4,c6,c9 are_collinear or not c2,c7,c8
  are_collinear or not c2,c6,c10 are_collinear or not c3,c7,c9
  are_collinear or not c3,c5,c10 are_collinear)) 
  proof
    assume that
A1: c1 <> c2 and
A2: c3 <> c1 and
A3: c2 <> c3 and
A4: c2 <> c4 and
A5: c3 <> c4 and
A6: c1 <> c5 and
A7: c1 <> c6 and
A8: c5 <> c6 and
A9: c5 <> c7 and
A10: c6 <> c7 and
A11: not c1,c4,c7 are_collinear and
A12: c1,c4,c2 are_collinear and
A13: c1,c4,c3 are_collinear and
A14: c1,c7,c5 are_collinear and
A15: c1,c7,c6 are_collinear and
A16: c4,c5,c8 are_collinear and
A17: c7,c2,c8 are_collinear and
A18: c4,c6,c9 are_collinear and
A19: c3,c7,c9 are_collinear and
A20: c2,c6,c10 are_collinear and
A21: c3,c5,c10 are_collinear;
    assume
A22: c4,c2,c7 are_collinear or c4,c3,c7 are_collinear or
    c2,c3,c7 are_collinear or c4,c2,c5 are_collinear or c4,c2,c6
    are_collinear or c4,c3,c5 are_collinear or c4,c3,c6 are_collinear or
    c2,c7,c5 are_collinear or c2,c7,c6 are_collinear or c3,c7,c5
    are_collinear or c3,c7,c6 are_collinear or c2,c3,c5 are_collinear or
    c2,c3,c6 are_collinear or c7,c5,c4 are_collinear or c7,c6,c4
    are_collinear or c5,c6,c4 are_collinear or c5,c6,c2 are_collinear or
    not c4,c5,c8 are_collinear or not c4,c6,c9 are_collinear or
    not c2,c7,c8 are_collinear or not c2,c6,c10 are_collinear or
    not c3,c7,c9 are_collinear or not c3,c5,c10 are_collinear;
    now
      thus not c1,c7,c4 are_collinear by COLLSP:4,A11;
      thus c4,c2,c1 are_collinear by A12,HESSENBE:1;
      thus c4,c3,c1 are_collinear by HESSENBE:1,A13;
      thus not c2,c3,c7 are_collinear by A3,A11,A12,A13,HESSENBE:3;
      thus c1,c5,c7 are_collinear by A14,COLLSP:4;
      thus c7,c1,c5 are_collinear by A14,COLLSP:4;
      thus c7,c6,c1 are_collinear by HESSENBE:1,A15;
      thus c1,c6,c7 are_collinear by A15,COLLSP:4;
      thus c7,c1,c6 are_collinear by A15,COLLSP:4;
      (for v100 holds v100,c4,c4 are_collinear) &
       (for v2,v100 holds not v100,c4,c4 are_collinear or
       not v100,c4,c3 are_collinear or not c4,c3,v2 are_collinear or
       v100,c4,v2 are_collinear) by COLLSP:2,A5,HESSENBE:3;
      hence not c4,c3,c7 are_collinear by A11,A13;
      (for v100 holds v100,c4,c4 are_collinear) &
       (for v2,v100 holds not v100,c4,c4 are_collinear or
       not v100,c4,c2 are_collinear or not c4,c2,v2 are_collinear or
      v100,c4,v2 are_collinear) by COLLSP:2,A4,HESSENBE:3;
      hence not c4,c2,c7 are_collinear by A11,A12;
      (for v100 holds v100,c1,c1 are_collinear) &
        (for v2,v100 holds not v100,c1,c1 are_collinear or
        not v100,c1,c2 are_collinear or not c1,c2,v2 are_collinear or
        v100,c1,v2 are_collinear) & c1,c2,c4 are_collinear &
      not c7,c1,c4 are_collinear by A12,COLLSP:2,A1,HESSENBE:1,3,A11;
      hence not c7,c1,c2 are_collinear;
      thus for v2,v100 holds not v100,c7,c7 are_collinear or
        not v100,c7,c6 are_collinear or  not c7,c6,v2 are_collinear or
        v100,c7,v2 are_collinear by A10,HESSENBE:3;
      thus for v100 holds v100,c7,c7 are_collinear by COLLSP:2;
      thus for v2,v100 holds not v100,c7,c7 are_collinear or
        not v100,c7,c5 are_collinear or not c7,c5,v2 are_collinear or
        v100,c7,v2  are_collinear by A9,HESSENBE:3;
      now
        thus c3,c7,c7 are_collinear by COLLSP:2;
        c1,c4,c1 are_collinear by COLLSP:2;
        hence not c3,c1,c7 are_collinear by A11,A2,A13,HESSENBE:3;
      end;
      hence not c3,c7,c1 are_collinear by HESSENBE:2,A11,A13;
      thus c5,c7,c1 are_collinear by HESSENBE:1,A14;
      now
        thus c2,c7,c7 are_collinear by COLLSP:2;
        c1,c4,c1 are_collinear by COLLSP:2;
        hence not c2,c1,c7 are_collinear by A11,A1,A12,HESSENBE:3;
      end;
      hence not c2,c7,c1 are_collinear by A11,A12,HESSENBE:2;
      not c4=c1 by COLLSP:2,A11;
      then c1,c2,c3 are_collinear by A13,A12,HESSENBE:2;
      hence c2,c3,c1 are_collinear by HESSENBE:1;
      hereby
        let v2,v100;
        not v100,c7,c5 are_collinear or not v100,c7,c7 are_collinear or
          not c5,c7,v2 are_collinear or v100,c7,v2 are_collinear
          by A9,HESSENBE:3;
        hence not v100,c7,c5 are_collinear or
        not c5,c7,v2 are_collinear or v100,c7,v2 are_collinear by COLLSP:2;
      end;
    end;
    hence contradiction
      by A6,HESSENBE:3,A17,A16,A18,A19,A20,A21,A22,COLLSP:4,A15,A8,A14,A7;
  end;
