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
  meet {B (-)X where X is binary-image of E: X in F} c= B(-)(meet F)
  proof
    per cases;
    suppose
      A1: F = {};
      reconsider Z = meet F as Subset of E;
      {B(-)X where X is binary-image of E: X in F} = {}
      proof
        assume { B(-)X where X is binary-image of E: X in F} <> {};
        then consider x be object such that
        A2: x in {B(-)X where X is binary-image of E: X in F}
        by XBOOLE_0:def 1;
        ex X be binary-image of E st x = B(-)X & X in F by A2;
        hence contradiction by A1;
      end;
      then {} = meet {B(-)X where X is binary-image of E: X in F}
      by SETFAM_1:def 1;
      hence meet {B(-)X where X is binary-image of E: X in F}
      c= B (-)(meet F);
    end;
    suppose F <> {}; then
      consider W0 be object such that
      A3: W0 in F by XBOOLE_0:def 1;
      reconsider W0 as binary-image of E by A3;
        let x be object;
        assume
        A4: x in meet { B(-)W where W is binary-image of E: W in F};
        A5: 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 A4,SETFAM_1:def 1;
        end;
        A6: x in B(-)W0 by A3,A5;
        then reconsider z=x as Element of E;
        for f be Element of E st f in meet F holds z - f in B
        proof
          let f be Element of E;
          assume
          A7: f in meet F;
          A8: meet F c= W0 by A3,SETFAM_1:3;
          consider zz be Element of E such that
          A9: z = zz & for w be Element of E st w in W0
          holds zz - w in B by A6;
          thus z - f in B by A9,A7,A8;
        end;
        hence x in B(-)(meet F);
    end;
  end;
