theorem Th04:
  r <> 0 & u is non zero implies r * u is non zero
  proof
    assume that
A1: r <> 0 and
A2: u is non zero;
    r * u <> 0.TOP-REAL 3
    proof
      assume
A3:   r * u = 0.TOP-REAL 3;
      u = 1 * u by RVSUM_1:52
       .= (1/r * r) * u by A1,XCMPLX_1:87
       .= 1/r * 0.TOP-REAL 3 by A3,RVSUM_1:49
       .= 0.TOP-REAL 3;
      hence contradiction by A2;
    end;
    hence thesis;
  end;
