
theorem Th03:
  for P being POINT of TarskiEuclid2Space st
  Tn2TR P is Element of inside_of_circle(0,0,1) holds BK_to_T2 T2_to_BK P = P
  proof
    let P be POINT of TarskiEuclid2Space;
    assume Tn2TR P is Element of inside_of_circle(0,0,1);
    then consider u be non zero Element of TOP-REAL 3 such that
A1: T2_to_BK P = Dir u and
A2: u`3 = 1 and
A3: Tn2TR P = |[u`1,u`2]| by Def02;
    reconsider Q9 = T2_to_BK P as Element of BK-model-Plane;
    reconsider Q = T2_to_BK P as Element of BK_model;
    consider p be Element of BK_model such that
A4: Q9 = p and
A5: BK_to_T2 Q9 = BK_to_REAL2 p by Def01;
    consider v be non zero Element of TOP-REAL 3 such that
A6: Dir v = p and
A7: v.3 = 1 and
A8: BK_to_REAL2 p = |[v.1,v.2]| by BKMODEL2:def 2;
    are_Prop u,v by A4,A6,A1,ANPROJ_1:22;
    then consider a be Real such that a <> 0 and
A9: u = a * v by ANPROJ_1:1;
A10: 1 = a * v`3 by A2,A9,EUCLID_5:9
     .= a * 1 by A7,EUCLID_5:def 3
     .= a;
    |[u`1,u`2,u`3]| = u by EUCLID_5:3
                   .= |[1 * v`1,1 * v`2,1 * v`3]| by A9,A10,EUCLID_5:7;
    then u`1 = v`1 & u`2 = v`2 by FINSEQ_1:78;
    then u`1 = v.1 & u`2 = v.2 by EUCLID_5:def 1,def 2;
    hence thesis by A5,A8,A3;
  end;
