
theorem
  for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj1 P = u.2/u.1 * normalize_proj2 P &
  normalize_proj2 P = u.1/u.2 * normalize_proj1 P
  proof
    let P be non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    set r = u.1 / u.2;
A2: u.1 <> 0 & u.2 <> 0 by A1,Th10,Th13;
A3: (u.1/u.2) * (u.2/u.1) = r * (1 / r) by XCMPLX_1:57
                         .= 1 by A2,XCMPLX_1:106;
    Dir normalize_proj1 P = P & Dir normalize_proj2 P = P by Def2,Def4;
    then are_Prop normalize_proj1 P,normalize_proj2 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A4: normalize_proj1 P = a * normalize_proj2 P by ANPROJ_1:1;
    normalize_proj1 P = |[1, u.2/u.1, u.3/u.1]| &
      normalize_proj2 P = |[u.1/u.2, 1, u.3/u.2]| by A1,Th11,Th14;
    then
A5: |[1, u.2/u.1, u.3/u.1]| = |[ a * (u.1 / u.2),a*1,a * (u.3/u.2)]|
      by A4,EUCLID_5:8;
    hence normalize_proj1 P = (u.2/u.1) * normalize_proj2 P by A4,FINSEQ_1:78;
    (u.1/u.2) * normalize_proj1 P
      = (u.1/u.2) * ((u.2/u.1) * normalize_proj2 P) by A4,A5,FINSEQ_1:78
     .= ((u.1/u.2) * (u.2/u.1)) * normalize_proj2 P by RVSUM_1:49
     .= normalize_proj2 P by A3,RVSUM_1:52;
    hence normalize_proj2 P = (u.1/u.2) * normalize_proj1 P;
  end;
