
theorem Th7:
  for u,v,w being non zero Element of TOP-REAL 3 st
  |(u,v)| = 0 & are_Prop w,v holds |(u,w)| = 0
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: |(u,v)| = 0 and
A2: are_Prop w,v;
    consider a be Real such that
    a <> 0 and
A3: w = a * v by A2,ANPROJ_1:1;
    reconsider un = u,vn = v as Element of REAL 3 by EUCLID:22;
    thus |(u,w)| = |(a * vn,un)| by A3
                .= a * |(v,u)| by EUCLID_8:68
                .= 0 by A1;
  end;
