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
  A(-)B = meet {b + A where b is Element of E: b in B}
  proof
    consider g be object such that
    A1: g in B by XBOOLE_0:def 1;
    reconsider g as Element of E by A1;
    A2: g+A in {b + A where b is Element of E: b in B} by A1;

    now let x be object;
      assume x in A(-)B; then
      consider z be Element of E such that
      A3: x = z & for b be Element of E st b in B
      holds z - b in A;

      for Y be set st Y in {b + A where b is Element of E: b in B}
      holds z in Y
      proof
        let Y be set;
        assume Y in {b + A where b is Element of E: b in B};
        then
        consider b be Element of E such that
        A4: Y = b + A & b in B;
        A5: z - b in A by A3,A4;
        z = b + (z- b) by RLVECT_4:1;
        hence z in Y by A5,A4;
      end;
      hence x in meet {b + A where b is Element of E: b in B}
      by A3,A2,SETFAM_1:def 1;
    end;
    hence A(-)B c= meet {b + A where b is Element of E: b in B};

    now let x be object;
      assume
      A6: x in meet {b + A where b is Element of E: b in B};
      consider S be set such that
      A7: S in {b + A where b is Element of E: b in B} by A2;
      consider d be Element of E such that
      A8: S = d + A & d in B by A7;

      x in S by A6,A7,SETFAM_1:def 1;
      then
      reconsider x0 = x as Element of E by A8;

      for b be Element of E st b in B holds x0 - b in A
      proof
        let b be Element of E;
        assume b in B;
        then b + A in {f + A where f is Element of E :f in B};
        then x in b + A by A6,SETFAM_1:def 1;
        then consider a be Element of E such that
        A9: x0 = b + a & a in A;

        thus thesis by A9,RLVECT_4:1;
      end;
      hence x in A(-)B;
    end;
    hence meet {b + A where b is Element of E: b in B} c= A(-)B;
  end;
