
theorem Th11:
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]|
  proof
    let P be non zero_proj1 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_proj1 P by Def2;
    then are_Prop u9,normalize_proj1 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj1 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj1 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj1 P)`1 by Def2
     .= a * u9`1 by A2;
    then
A4: u9`1 = 1 / a & a = 1 / u9`1 by XCMPLX_1:73;
    normalize_proj1 P = |[ 1,u9`2 / u9`1,(1 / u9`1) * u9`3]|
                         by A1,A3,A4,EUCLID_5:7
                     .= |[ 1,u9.2 / u9.1,u9.3/u9.1]|;
    hence thesis;
  end;
