
theorem
  for P being non point_at_infty Element of ProjectiveSpace TOP-REAL 3
  st P in BK_model holds RP3_to_REAL2 P in inside_of_circle(0,0,1)
  proof
    let P be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume P in BK_model;
    then reconsider P1 = P as Element of BK_model;
    consider u be non zero Element of TOP-REAL 3 such that
A1: Dir u = P1 & u.3 = 1 and
A2: BK_to_REAL2 P1 = |[u.1,u.2]| by BKMODEL2:def 2;
    |[u`1,u.2]| is Element of inside_of_circle(0,0,1) by A2,EUCLID_5:def 1;
    then
A3: |[u`1,u`2]| is Element of inside_of_circle(0,0,1) by EUCLID_5:def 2;
    consider v be non zero Element of TOP-REAL 3 such that
A4: P = Dir v & v`3 = 1 & RP3_to_REAL2 P = |[v`1,v`2]| by Def05;
    Dir v = Dir u & u.3 <> 0 & u.3 = v.3 by A1,A4,EUCLID_5:def 3;
    then u = v by Th16;
    hence thesis by A4,A3;
  end;
