reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem Th23:
  for u,v,w being non zero Element of TOP-REAL 3 st
  u`1 <> 0 & u`3 = 1 & v`1 = - u`2 & v`2 = u`1 & v`3 = 0 &
  w`3 = 1 & |{ u,v,w }| = 0 & 1 < (u`1)^2 + (u`2)^2 holds
  (w`1)^2 + (w`2)^2 <> 1
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A0: u`1 <> 0 and
A1: u`3 = 1 and
A2: v`1 = - u`2 and
A3: v`2 = u`1 and
A4: v`3 = 0 and
A5: w`3 = 1 and
A6: |{ u,v,w }| = 0 and
A7: 1 < (u`1)^2 + (u`2)^2;
A8:  (u`1)^2 + (u`2)^2 - (u`1) * (w`1) - (u`2) * (w`2) = 0
      by A1,A2,A3,A4,A5,A6,Th21;
    assume
A9: (w`1)^2 + (w`2)^2 = 1;
    reconsider x = w`1, y = w`2 as Real;
    reconsider u1 = u`1 as non zero Real by A0;
    reconsider r = sqrt ((u`1)^2 + (u`2)^2) as positive Real by A0,Th02;
    reconsider u2 = u`2 as Real;
A10: r^2 = (u`1)^2 + (u`2)^2 by SQUARE_1:def 2;
    then u1 * x = r^2 - u2 * y by A8;
    then ((r^2)^2 / u1^2) - 2 * ((r^2 * u2) / (u1 * u1)) * y
      + (u2^2 / u1^2) * y^2 + y^2 = 1 by A9,Th03;
    hence contradiction by A10,A7,Th09;
  end;
