reserve E for RealLinearSpace;
reserve A, B, C for binary-image of E;
reserve a, b, v for Element of E;
reserve F, G for binary-image-family of E;
reserve A, B, C for non empty binary-image of E;

theorem Th7:
  A(+)B = {v where v is Element of E: (v + (-1)*B) /\ A <> {}}
  proof
    thus A(+)B c= {v where v is Element of E: (v + (-1)*B ) /\ A <> {}}
    proof let x be object;
      assume
      A1: x in A(+)B;
      consider a0,b0 be Element of E such that
      A2: x = a0 + b0 & a0 in A & b0 in B by A1;
      reconsider v = x as Element of E by A1;
      A3: v - b0 = a0 by A2,RLVECT_4:1;
      (-1)*b0 in (-1)*B by A2;
      then v + (-1)*b0 in v + (-1)*B;
      then v - b0 in v + (-1)*B by RLVECT_1:16;
      then (v + (-1)*B) /\ A <> {} by A2,A3,XBOOLE_0:def 4;
      hence x in {w where w is Element of E: (w + (-1)*B) /\ A <> {}};
    end;

    let x be object;
      assume x in {v where v is Element of E: (v + (-1)*B) /\ A <> {}};
      then
      consider v be Element of E such that
      A4: x = v & (v + (-1)*B ) /\ A <> {};
      consider y be object such that
      A5: y in ((v + (-1)*B) /\ A) by A4,XBOOLE_0:def 1;
      A6: y in (v + (-1)*B ) & y in A by A5,XBOOLE_0:def 4;
      then
      consider nb be Element of E such that
      A7: y = v + nb & nb in (-1)*B;
      consider b be Element of E such that
      A8: nb = (-1)*b & b in B by A7;
      reconsider a = y as Element of E by A7;
      a + b = v - b + b by A7,A8,RLVECT_1:16
      .= v by RLVECT_4:1;
      hence x in A(+)B by A4,A8,A6;
  end;
