
theorem Th9:
  for u,v being non zero Element of TOP-REAL 3
  for r being Real st r <> 0 & are_Prop u,v holds are_Prop r * u,v
  proof
    let u,v be non zero Element of TOP-REAL 3;
    let r be Real;
    assume that
A1: r <> 0 and
A2: are_Prop u,v;
    consider a be Real such that
A3: a <> 0 and
A4: u = a * v by ANPROJ_1:1,A2;
    r * u = (r * a) * v by A4,RVSUM_1:49;
    hence thesis by A1,A3,ANPROJ_1:1;
  end;
