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
  F <> {}
  implies B(-) (union F) = meet {B(-)X where X is binary-image of E: X in F}
  proof
    assume F <> {};
    then consider W0 be object such that
    A1: W0 in F by XBOOLE_0:def 1;
    reconsider W0 as binary-image of E by A1;
    A2: B(-)W0 in {B(-)X where X is binary-image of E: X in F} by A1;
    for x be object holds x in B(-) (union F)
    iff x in meet {B(-)X where X is binary-image of E: X in F}
    proof
      let x be object;
      hereby
        assume x in B(-) (union F);
        then consider z be Element of E such that
        A3: x = z & for f be Element of E st f in (union F) holds z - f in B;
        now
          let Y be set;
          assume Y in {B(-)X where X is binary-image of E: X in F};
          then consider X be binary-image of E such that
          A4: Y = B(-)X & X in F;
          now
            let f be Element of E;
            assume f in X;
            then f in (union F) by A4,TARSKI:def 4;
            hence z - f in B by A3;
          end;
          hence x in Y by A3,A4;
        end;
        hence x in meet {B(-)W where W is binary-image of E: W in F}
        by A2,SETFAM_1:def 1;
      end;
      assume
      A5: x in meet {B(-)W where W is binary-image of E: W in F};
      A6: for W be binary-image of E st W in F holds x in B(-)W
      proof
        let W be binary-image of E;
        assume W in F;
        then B(-)W in {B(-)D where D is binary-image of E: D in F};
        hence x in B(-)W by A5,SETFAM_1:def 1;
      end;
      x in B(-)W0 by A1,A6;
      then reconsider z=x as Element of E;
      for f be Element of E st f in (union F) holds z - f in B
      proof
        let f be Element of E;
        assume f in (union F);
        then consider W be set such that
        A7: f in W & W in F by TARSKI:def 4;
        reconsider W as binary-image of E by A7;
        z in B(-)W by A6,A7; then
        consider w be Element of E such that
        A8: z = w & for f be Element of E st f in W holds w - f in B;
        thus z - f in B by A7,A8;
      end;
      hence x in B(-)(union F);
    end;
    hence B(-)(union F) = meet {B(-)X where X is binary-image of E: X in F}
    by TARSKI:2;
  end;
