
theorem Th17:
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]|
  proof
    let P be non zero_proj3 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_proj3 P by Def6;
    then are_Prop u9,normalize_proj3 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj3 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj3 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj3 P)`3 by Def6
     .= a * u9`3 by A2;
    then
A4: u9`3 = 1 / a & a = 1 / u9`3 by XCMPLX_1:73;
    normalize_proj3 P = |[ u9`1 / u9`3,(1 / u9`3) * u9`2,1]|
                        by A1,A3,A4,EUCLID_5:7
                     .= |[ u9.1 / u9.3,u9.2/u9.3,1]|;
    hence thesis;
  end;
