
theorem Th63:
  for P being Element of BK_model
  for P9 being non point_at_infty Element of ProjectiveSpace TOP-REAL 3 st
  P = P9 holds RP3_to_REAL2 P9 = BK_to_REAL2 P
  proof
    let P be Element of BK_model;
    let P9 be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume
A1: P = P9;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P9 = Dir u & u`3 = 1 & RP3_to_REAL2 P9 = |[u`1,u`2]| by Def05;
    consider v be non zero Element of TOP-REAL 3 such that
A3: Dir v = P & v.3 = 1 & BK_to_REAL2 P = |[v.1,v.2]| by BKMODEL2:def 2;
    Dir u = Dir v & u.3 = v.3 & u.3 <> 0 by A1,A2,A3,EUCLID_5:def 3;
    then u = v by Th16;
    then u`1 = v.1 & u`2 = v.2 by EUCLID_5:def 1,def 2;
    hence thesis by A2,A3;
  end;
