theorem Th43:
  aa <> 0.RR implies Carrier(LL * aa) = Carrier(LL)
proof
  set T = {uu : (LL * aa).uu <> 0.RR};
  set S = {vv : LL.vv <> 0.RR};
  assume
A1: aa <> 0.RR;
  T = S
  proof
    thus T c= S
    proof
      let x be object;
      assume x in T;
      then consider uu such that
A2:   x = uu and
A3:   (LL * aa).uu <> 0.RR;
      (LL * aa).uu = LL.uu * aa by Def10;
      then LL.uu <> 0.RR by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider vv such that
A4: x = vv and
A5: LL.vv <> 0.RR;
    (LL * aa).vv = LL.vv * aa by Def10;
    then (LL * aa).vv <> 0.RR by A1,A5,VECTSP_2:def 1;
    hence thesis by A4;
  end;
  hence thesis;
end;
