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

theorem Th2:
  for E be RealLinearSpace, A, B be Subset of E st A <> {} & B = {}
  holds B(-)A = B
  proof
    let E be RealLinearSpace,
    A, B be Subset of E;
    assume
    A1: A <> {} & B = {};
    then consider a be object such that
    A2: a in A by XBOOLE_0:def 1;
    reconsider a as Element of E by A2;
    assume B(-)A <> B;
    then consider ba be object such that
    A3: ba in B(-)A by A1,XBOOLE_0:def 1;
    consider z be Element of E such that
    A4: ba = z & for a be Element of E st a in A holds z - a in B by A3;
    thus contradiction by A1,A2,A4;
  end;
