reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem Th66:
  for O being Matrix of 3,REAL
  for P being Element of ProjectiveSpace TOP-REAL 3
  for p being FinSequence of REAL st
  O = symmetric_3(1,1,-1,0,0,0) & P = Dir u & u = p
  holds P in absolute iff SumAll QuadraticForm(p,O,p) = 0
  proof
    let O be Matrix of 3,REAL;
    let P be Element of ProjectiveSpace TOP-REAL 3;
    let p be FinSequence of REAL;
    assume
A1: O = symmetric_3(1,1,-1,0,0,0) & P = Dir u & u = p;
    hereby
      assume P in absolute;
      then P 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) = 0} by PASCAL:def 2;
      then consider Q being Point of ProjectiveSpace TOP-REAL 3 such that
A2:   P = Q and
A3:   for u being Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
        qfconic(1,1,-1,0,0,0,u) = 0;
      consider u1 be Element of TOP-REAL 3 such that
A4:   u1 is non zero and
A5:   Q = Dir u1 by ANPROJ_1:26;
      reconsider p1 = u1 as FinSequence of REAL by EUCLID:24;
A6:   SumAll QuadraticForm(p1,O,p1) = qfconic(1,1,-1,2 * 0,2 * 0, 2 * 0,u1)
                                       by A1,PASCAL:13
                                   .= 0 by A3,A4,A5;
      are_Prop u1,u by A2,A5,A1,A4,ANPROJ_1:22;
      then consider a be Real such that
A7:   a <> 0 and
A8:   u1 = a * u by ANPROJ_1:1;
      reconsider fa = a as Element of F_Real by XREAL_0:def 1;
      u is Element of REAL 3 by EUCLID:22;
      then len p = 3 by A1,EUCLID_8:50;
      then SumAll QuadraticForm(p1,O,p1)
        = fa * fa * SumAll QuadraticForm(p,O,p) by A1,A8,Th35;
      hence SumAll QuadraticForm(p,O,p) = 0 by A7,A6;
    end;
    assume SumAll QuadraticForm(p,O,p) = 0;
    then 0 = qfconic(1,1,-1,2 * 0,2 * 0, 2 * 0,u) by A1,PASCAL:13
          .= qfconic(1,1,-1,0,0, 0,u);
    then 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) = 0 by A1,PASCAL:10;
    then P 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) = 0};
    hence P in absolute by PASCAL:def 2;
  end;
