theorem Th52:
  Carrier(- L) = Carrier(L)
proof
  set a = -1_R;
  Carrier(L * a) = Carrier(L)
  proof
    set S = {v : L.v <> 0.R};
    set T = {u : (L * a).u <> 0.R};
    T = S
    proof
      thus T c= S
      proof
        let x be object;
        assume x in T;
        then consider u such that
A1:     x = u and
A2:     (L * a).u <> 0.R;
        (L * a).u = L.u * a by Def10;
        then L.u <> 0.R by A2;
        hence thesis by A1;
      end;
      let x be object;
      assume x in S;
      then consider v such that
A3:   x = v and
A4:   L.v <> 0.R;
      (L * a).v = L.v * a by Def10
        .= -(L.v) by VECTSP_2:28;
      then (L * a).v <> 0.R by A4,VECTSP_2:3;
      hence thesis by A3;
    end;
    hence thesis;
  end;
  hence thesis;
end;
