
theorem Th11:
  for P being Element of real_projective_plane holds
  ex Q being Element of BK_model st P <> Q
  proof
    let P be Element of real_projective_plane;
    per cases;
    suppose
A1:   #P = |[0,0,1]|;
      reconsider u = |[0,1/2,1]| as non zero Element of TOP-REAL 3;
      reconsider Q = Dir u as Point of ProjectiveSpace TOP-REAL 3
        by ANPROJ_1:26;
      now
        let v be Element of TOP-REAL 3;
        assume v is non zero & Q = Dir v;
        then are_Prop u,v by ANPROJ_1:22;
        then consider a be Real such that
A2:     a <> 0 and
A3:     v = a * u by ANPROJ_1:1;
        v = |[a * 0,a * (1/2),a * 1]| by A3,EUCLID_5:8
         .= |[0,a/2,a]|; then
A4:     v.1 = 0 & v.2 = a/2 & v.3 = a by FINSEQ_1:45;
        qfconic(1,1,-1,0,0,0,v) = 1 * v.1 * v.1 + 1 * v.2 * v.2
          + (- 1) * v.3 * v.3
          + 0 * v.1 * v.2 + 0 * v.1 * v.3 + 0 * v.2 * v.3 by PASCAL:def 1
                               .= a^2 * (-3/4) by A4;
        hence qfconic(1,1,-1,0,0,0,v) is negative by A2;
      end;
      then Q in {P where P is Point of ProjectiveSpace TOP-REAL 3:
        for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
        qfconic(1,1,-1,0,0,0,u) is negative};
      then reconsider Q as Element of BK_model
        by BKMODEL2:def 1;
      reconsider Q9 = Q as Element of real_projective_plane;
      take Q;
      Q <> P
      proof
        assume
A6:     Q = P;
A7:     Q9 = Dir u & u.3 = 1 by FINSEQ_1:45;
        Q9 in absolute \/ BK_model by XBOOLE_0:def 3;
        then |[0,0,1]| = |[0,1/2,1]| by A7,Def01,A6,A1;
        hence contradiction by FINSEQ_1:78;
      end;
      hence thesis;
    end;
    suppose
A8:   #P <> |[0,0,1]|;
      reconsider u = |[0,0,1]| as non zero Element of TOP-REAL 3;
      reconsider Q = Dir u as Point of ProjectiveSpace TOP-REAL 3
        by ANPROJ_1:26;
      now
        let v be Element of TOP-REAL 3;
        assume v is non zero & Q = Dir v;
        then are_Prop u,v by ANPROJ_1:22;
        then consider a be Real such that
A9:     a <> 0 and
A10:    v = a * u by ANPROJ_1:1;
        v = |[a * 0,a * 0,a * 1]| by A10,EUCLID_5:8
         .= |[0,0,a]|; then
A11:    v.1 = 0 & v.2 = 0 & v.3 = a by FINSEQ_1:45;
        qfconic(1,1,-1,0,0,0,v) = 1 * v.1 * v.1 + 1 * v.2 * v.2
          + (- 1) * v.3 * v.3
          + 0 * v.1 * v.2 + 0 * v.1 * v.3 + 0 * v.2 * v.3 by PASCAL:def 1
                               .= a^2 * (-1) by A11;
        hence qfconic(1,1,-1,0,0,0,v) is negative by A9;
      end;
      then Q in {P where P is Point of ProjectiveSpace TOP-REAL 3:
        for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
        qfconic(1,1,-1,0,0,0,u) is negative};
      then reconsider Q as Element of BK_model
        by BKMODEL2:def 1;
      reconsider Q9 = Q as Element of real_projective_plane;
      take Q;
      Q <> P
      proof
        assume
A13:    Q = P;
A14:    Q9 = Dir u & u.3 = 1 by FINSEQ_1:45;
        Q9 in absolute \/ BK_model by XBOOLE_0:def 3;
        hence contradiction by A14,A8,Def01,A13;
      end;
      hence thesis;
    end;
  end;
