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;
reserve P1,P2,P3,P4,P5,P6,P7,P8,P9 for Point of ProjectiveSpace TOP-REAL 3,
                       a,b,c,d,e,f for Real;

theorem Th32:
  P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration
  implies
  not P7,P2,P5 are_collinear & not P1,P5,P7 are_collinear &
  not P2,P4,P7 are_collinear & not P2,P5,P9 are_collinear &
  not P2,P6,P8 are_collinear & not P2,P7,P8 are_collinear &
  not P2,P7,P9 are_collinear & not P3,P5,P8 are_collinear &
  not P3,P6,P8 are_collinear & not P5,P7,P8 are_collinear &
  not P5,P7,P9 are_collinear
  proof
    assume
A1: P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration;
    assume
A2: not(
      not P7,P2,P5 are_collinear & not P1,P5,P7 are_collinear &
      not P2,P4,P7 are_collinear & not P2,P5,P9 are_collinear &
      not P2,P6,P8 are_collinear & not P2,P7,P8 are_collinear &
      not P2,P7,P9 are_collinear & not P3,P5,P8 are_collinear &
      not P3,P6,P8 are_collinear & not P5,P7,P8 are_collinear &
      not P5,P7,P9 are_collinear);
    not P1,P2,P4 are_collinear & not P1,P2,P5 are_collinear &
    not P1,P2,P6 are_collinear & not P1,P3,P4 are_collinear &
    not P1,P3,P5 are_collinear & not P1,P3,P6 are_collinear &
    not P2,P3,P4 are_collinear & not P2,P3,P5 are_collinear &
    not P4,P5,P1 are_collinear & not P4,P5,P2 are_collinear &
    not P4,P5,P3 are_collinear & not P4,P6,P2 are_collinear &
    not P4,P6,P3 are_collinear & not P5,P6,P2 are_collinear &
    P1,P5,P9 are_collinear & P1,P6,P8 are_collinear &
    P2,P4,P9 are_collinear & P2,P6,P7 are_collinear &
    P3,P4,P8 are_collinear & P3,P5,P7 are_collinear
      by A1,ANPROJ_8:57,HESSENBE:1;
    hence contradiction by A2,ANPROJ_8:57,Th31;
  end;
