
theorem Th02:
  for P,Q being Element of BK_model holds P = Q iff
  BK_to_REAL2 P = BK_to_REAL2 Q
  proof
    let P,Q be Element of BK_model;
    thus P = Q implies BK_to_REAL2 P = BK_to_REAL2 Q;
    assume
A1: BK_to_REAL2 P = BK_to_REAL2 Q;
    consider u be non zero Element of TOP-REAL 3 such that
A2: Dir u = P & u.3 = 1 & BK_to_REAL2 P = |[u.1,u.2]| by Def01;
    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 Def01;
    u.1 = v.1 & u.2 = v.2 & u.3 = v.3 by A1,A2,A3,FINSEQ_1:77;
    then u`1 = v.1 & u`2 = v.2 & u`3 = v.3 by EUCLID_5:def 1,def 2,def 3; then
A4: u`1 = v`1 & u`2 = v`2 & u`3 = v`3 by EUCLID_5:def 1,def 2,def 3;
    u = |[u`1,u`2,u`3]| & v = |[v`1,v`2,v`3]| by EUCLID_5:3;
    hence thesis by A2,A3,A4;
  end;
