reserve E for RealLinearSpace;
reserve A, B, C for binary-image of E;
reserve a, b, v for Element of E;

theorem Th3:
  for E be RealLinearSpace,
  A, B be Subset of E st B = the carrier of E & A <> {}
  holds A(+)B = B & B(+)A = B
  proof
    let E be RealLinearSpace, A, B be Subset of E;
    assume
    A1: B = the carrier of E & A <> {};
    then consider a be object such that
    A2: a in A by XBOOLE_0:def 1;
    reconsider a as Element of E by A2;
    now let x be object;
      assume x in the carrier of E;
      then reconsider z = x as Element of E;
      a+(z-a) =z by RLVECT_4:1;
      hence x in A(+)B by A1,A2;
    end;
    then
    A3: the carrier of E c= A(+)B;
    hence B = A(+)B by A1;
    thus B(+)A = A + B
    .= B by A3,A1;
  end;
