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

theorem Th4:
  for E be RealLinearSpace,
  A, B be Subset of E st B = the carrier of E
  holds B(-)A = B
  proof
    let E be RealLinearSpace, A,B be Subset of E;
    assume
    A1: B = the carrier of E;
    now let x be object;
      assume x in the carrier of E;
      then reconsider z = x as Element of E;
      for a be Element of E st a in A holds z-a in B by A1;
      hence x in B(-)A;
    end;
    then the carrier of E c= B(-)A;
    hence B = B(-)A by A1;
  end;
