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

theorem Th1:
  for E be RealLinearSpace, A, B be Subset of E st B = {}
  holds A(+)B = B & B(+)A = B & A(-)B = the carrier of E
  proof
    let E be RealLinearSpace,
    A, B be Subset of E;
    assume A1: B = {};
    hence A(+)B = B & B(+)A = B by RUSUB_5:5;
    now let x be object;
      assume x in the carrier of E;
      then reconsider z = x as Element of E;
      for b be Element of E st b in B
      holds z - b in A by A1;
      hence x in A(-)B;
    end;
    then the carrier of E c= A(-)B;
    hence the carrier of E = A(-)B;
  end;
