
theorem Th54:
  for P being non point_at_infty Point of ProjectiveSpace TOP-REAL 3
  for Q being Element of BK_model st P = Q holds RP3_to_REAL2 P = BK_to_REAL2 Q
  proof
    let P be non point_at_infty Point of ProjectiveSpace TOP-REAL 3;
    let Q be Element of BK_model;
    assume
A1: P = Q;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P = Dir u & u`3 = 1 & RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
    consider v be non zero Element of TOP-REAL 3 such that
A3: Dir v = Q & v.3 = 1 & BK_to_REAL2 Q = |[v.1,v.2]| by BKMODEL2:def 2;
    Dir v = Dir u & u.3 <> 0 & u.3 = v.3 by A1,A2,A3,EUCLID_5:def 3;
    then u = v by Th16;
    then BK_to_REAL2 Q = |[u`1,u.2]| by A3,EUCLID_5:def 1
                      .= |[u`1,u`2]| by EUCLID_5:def 2;
    hence thesis by A2;
  end;
