
theorem Th16:
  for u,v being non zero Element of TOP-REAL 3 st
  Dir u = Dir v & u.3 <> 0 & u.3 = v.3 holds u = v
  proof
    let u,v be non zero Element of TOP-REAL 3;
    assume that
A1: Dir u = Dir v and
A2: u.3 <> 0 and
A3: u.3 = v.3;
    are_Prop u,v by A1,ANPROJ_1:22;
    then consider a be Real such that a <> 0 and
A4: u = a * v by ANPROJ_1:1;
    u.3 = a * u.3 by A3,A4,RVSUM_1:44;
    then a = 1 by A2,XCMPLX_1:7;
    hence thesis by A4,RVSUM_1:52;
  end;
