
theorem Th14:
  for P being 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_proj2 P = |[u.1/u.2,1,u.3/u.2]|
  proof
    let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u9 be non zero Element of TOP-REAL 3;
    assume P = Dir u9;
    then Dir u9 = Dir normalize_proj2 P by Def4;
    then are_Prop u9,normalize_proj2 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj2 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj2 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj2 P)`2 by Def4
     .= a * u9`2 by A2;
    then
A4: u9`2 = 1 / a & a = 1 / u9`2 by XCMPLX_1:73;
    normalize_proj2 P = |[ u9`1 / u9`2,1,(1 / u9`2) * u9`3]|
                         by A1,A3,A4,EUCLID_5:7
                     .= |[ u9.1 / u9.2,1,u9.3/u9.2]|;
    hence thesis;
  end;
