 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;
 reserve N,N1,N2 for invertible Matrix of 3,F_Real;
 reserve P,P1,P2,P3 for Point of ProjectiveSpace TOP-REAL 3;

theorem Th23:
  for P being Point of ProjectiveSpace TOP-REAL 3 st
  not (Dir100,Dir010,P are_collinear) &
  not (Dir100,Dir001,P are_collinear) &
  not (Dir010,Dir001,P are_collinear) holds
  ex a,b,c being non zero Element of F_Real st P = Dir |[a,b,c]|
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    assume that
A1: not (Dir100,Dir010,P are_collinear) and
A2: not (Dir100,Dir001,P are_collinear) and
A3: not (Dir010,Dir001,P are_collinear);
    consider a,b,c being Element of F_Real such that
A4: P = Dir |[a,b,c]| and
A5: (a <> 0 or b <> 0 or c <> 0) by Th22;
    a is non zero & b is non zero & c is non zero by A1,A2,A3,A4,A5,Lem07;
    then reconsider a,b,c as non zero Element of F_Real;
    take a,b,c;
    thus thesis by A4;
  end;
