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 Th21:
  for u,v,w being non zero Element of TOP-REAL 3 st
  u`3 = 1 & v`1 = - u`2 & v`2 = u`1 & v`3 = 0 &
  w`3 = 1 & |{ u,v,w }| = 0 holds
  (u`1)^2 + (u`2)^2 - (u`1) * (w`1) - (u`2) * (w`2) = 0
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
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;
    set p = u,q = v, r = w;
    0 = p`1 * q`2 * r`3 - 1 *q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1
        - p`2*q`1*r`3 +  1 *q`1*r`2 by A1,A6,ANPROJ_8:27
     .= (p`1)^2 + (p`2)^2 - p`1 * r`1 - p`2 * r`2 by A2,A3,A4,A5;
    hence thesis;
  end;
