
theorem Th17:
  for P being Element of BK_model
  for u being non zero Element of TOP-REAL 3 st
  P = Dir u & u.3 = 1 holds (u.1)^2 + (u.2)^2 < 1
  proof
    let P be Element of BK_model;
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u and
A2: u.3 = 1;
    consider u2 be non zero Element of TOP-REAL 3 such that
A3: Dir u2 = P & u2.3 = 1 & BK_to_REAL2 P = |[u2.1,u2.2]| by Def01;
A4: u = u2 by A1,A2,A3,BKMODEL1:43;
    reconsider S = |[u.1,u.2]| as Element of TOP-REAL 2;
A5: |. S - |[0,0]| .| = |. |[S`1,S`2]| - |[0,0]| .| by EUCLID:53
                          .= |. |[S`1 - 0,S`2 - 0]|  .| by EUCLID:62
                          .= |. S .| by EUCLID:53;
    1^2 = 1;
    then |. S .|^2 < 1 by A4,A3,TOPREAL9:45,A5,SQUARE_1:16;
    then (S`1)^2 + (S`2)^2 < 1 by JGRAPH_3:1;
    then (u.1)^2 + (S`2)^2 < 1 by EUCLID:52;
    hence thesis by EUCLID:52;
  end;
