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

theorem
  A(+)B = union {b + A where b is Element of E: b in B}
  proof
    now 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;
      A3: x in b0 + A by A2;
      b0 + A in {b + A where b is Element of E: b in B} by A2;
      hence x in union {b + A where b is Element of E: b in B}
      by A3,TARSKI:def 4;
    end;
    hence
    A(+)B c= union {b + A where b is Element of E: b in B};

    now let x be object;
      assume x in union {b + A where b is Element of E: b in B};
      then consider y be set such that
      A4: x in y & y in {b + A where b is Element of E: b in B}
      by TARSKI:def 4;
      consider b be Element of E such that
      A5: y = b + A & b in B by A4;
      consider a be Element of E such that
      A6: x = b + a & a in A by A5,A4;
      thus x in A(+)B by A5,A6;
    end;
    hence thesis;
  end;
