
theorem
  for P being Element of BK_model holds REAL2_to_BK BK_to_REAL2 P = P
  proof
    let P be Element of BK_model;
    consider u be non zero Element of TOP-REAL 3 such that
A1: Dir u = P and
A2: u.3 = 1 and
A3: BK_to_REAL2 P = |[u.1,u.2]| by Def01;
    consider Q be Element of TOP-REAL 2 such that
A4: Q = BK_to_REAL2 P and
A5: REAL2_to_BK BK_to_REAL2 P = Dir |[Q`1,Q`2,1]| by Def02;
A6: |[Q`1,Q`2,1]| is non zero by EUCLID_5:4,FINSEQ_1:78;
    are_Prop |[Q`1,Q`2,1]|,u
    proof
A7:   Q`1 = u.1 & Q`2 = u.2 by A3,A4,EUCLID:52;
      u = |[u`1,u`2,u`3]| by EUCLID_5:3
       .= |[u.1,u`2,u`3]| by EUCLID_5:def 1
       .= |[u.1,u.2,u`3]| by EUCLID_5:def 2
       .= |[u.1,u.2,u.3]| by EUCLID_5:def 3;
      hence thesis by A2,A7;
    end;
    hence thesis by A1,A5,A6,ANPROJ_1:22;
  end;
