
theorem
  for P being POINT of BK-model-Plane holds T2_to_BK BK_to_T2 P = P
  proof
    let P be POINT of BK-model-Plane;
    consider p be Element of BK_model such that
A1: P = p and
A2: BK_to_T2 P = BK_to_REAL2 p by Def01;
    consider u be non zero Element of TOP-REAL 3 such that
A3: Dir u = p and
A4: u.3 = 1 and
A5: BK_to_REAL2 p = |[u.1,u.2]| by BKMODEL2:def 2;
    reconsider Q = BK_to_T2 P as POINT of TarskiEuclid2Space;
    consider v be non zero Element of TOP-REAL 3 such that
A6: T2_to_BK Q = Dir v and
A7: v`3 = 1 and
A8: Tn2TR Q = |[v`1,v`2]| by A2,Def02;
    u.1 = v`1 & u.2 = v`2 by A8,A5,A2,FINSEQ_1:77; then
A9: u`1 = v`1 & u`2 = v`2 by EUCLID_5:def 1,def 2;
    u = |[u`1,u`2,u`3]| by EUCLID_5:3
     .= |[v`1,v`2,v`3]| by A4,A7,A9,EUCLID_5:def 3
     .= v by EUCLID_5:3;
    hence thesis by A1,A6,A3;
  end;
