
theorem
  for Q being Element of inside_of_circle(0,0,1) holds
  BK_to_REAL2 REAL2_to_BK Q = Q
  proof
    let Q be Element of inside_of_circle(0,0,1);
    consider P be Element of TOP-REAL 2 such that
A1: P = Q and
A2: REAL2_to_BK Q = Dir |[P`1,P`2,1]| by Def02;
    consider u be non zero Element of TOP-REAL 3 such that
A3: Dir u = REAL2_to_BK Q and
A4: u.3 = 1 and
A5: BK_to_REAL2 REAL2_to_BK Q = |[u.1,u.2]| by Def01;
    |[P`1,P`2,1]| is non zero by EUCLID_5:4,FINSEQ_1:78;
    then are_Prop |[P`1,P`2,1]|,u by A2,A3,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A6: |[P`1,P`2,1]| = a * u by ANPROJ_1:1;
A7: a = a * u.3 by A4
     .= (a * u).3 by RVSUM_1:44
     .= |[P`1,P`2,1]|`3 by A6,EUCLID_5:def 3
     .= 1 by EUCLID_5:2;
A8: |[P`1,P`2,1]| = u by A7,RVSUM_1:52,A6; then
A9: P`1 = u`1 by EUCLID_5:2
       .= u.1 by EUCLID_5:def 1;
    P`2 = u`2 by A8,EUCLID_5:2
       .= u.2 by EUCLID_5:def 2;
    hence thesis by A9,A5,A1,EUCLID:53;
  end;
