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 Th18:
  (meet F)(-)B = meet { X(-)B where X is binary-image of E: X in F}
  proof
    per cases;
    suppose
      A1: F = {};
      reconsider Z=(meet F) as Subset of E;
      A2: (meet F) = {} by A1,SETFAM_1:def 1;
      {X(-)B where X is binary-image of E: X in F} = {}
      proof
        assume {X(-)B where X is binary-image of E: X in F} <> {};
        then consider x be object such that
        A3: x in {X(-)B where X is binary-image of E: X in F}
        by XBOOLE_0:def 1;
        ex X be binary-image of E st x = X(-)B & X in F by A3;
        hence contradiction by A1;
      end;
      then {} = meet {X(-)B where X is binary-image of E: X in F}
      by SETFAM_1:def 1;
      hence (meet F)(-)B = meet {X(-)B where X is binary-image of E: X in F}
      by A2,Th2;
    end;

    suppose
      A4: F <> {};
      then consider W0 be object such that
      A5: W0 in F by XBOOLE_0:def 1;
      reconsider W0 as binary-image of E by A5;
      A6: W0(-)B in { W(-)B where W is binary-image of E: W in F} by A5;
      for x be object holds x in (meet F)(-)B
      iff x in meet {W(-)B where W is binary-image of E: W in F}
      proof
        let x be object;
        hereby
          assume x in (meet F)(-)B;
          then consider z be Element of E such that
          A7: x = z & for b be Element of E st b in B holds z - b in (meet F);
          now let Y be set;
            assume Y in { X(-)B where X is binary-image of E: X in F};
            then consider X be binary-image of E such that
            A8: Y = X(-)B & X in F;
            now let b be Element of E;
              assume b in B;
              then
              A9: z - b in (meet F) by A7;
              meet F c= X by A8,SETFAM_1:3;
              hence z - b in X  by A9;
            end;
            hence x in Y by A8,A7;
          end;
          hence x in meet {W(-)B where W is binary-image of E: W in F}
          by A6,SETFAM_1:def 1;
        end;
        assume
        A10: x in meet {W(-)B where W is binary-image of E: W in F};
        A11: for W be binary-image of E st W in F holds x in W(-)B
        proof
          let W be binary-image of E;
          assume W in F;
          then W(-)B in {D(-)B where D is binary-image of E: D in F};
          hence x in W(-)B by A10,SETFAM_1:def 1;
        end;

        x in W0(-)B by A5,A11;
        then
        reconsider z=x as Element of E;

        for b be Element of E st b in B holds z - b in meet F
        proof
          assume not for b be Element of E st b in B holds z - b in meet F;
          then consider b be Element of E such that
          A12: b in B & not z - b in meet F;
          consider X be set such that
          A13:  X in F & not (z-b) in X by A4,A12,SETFAM_1:def 1;
          reconsider X as binary-image of E by A13;
          z in X(-)B by A13,A11;
          then consider zz be Element of E such that
          A14: z = zz & for b be Element of E st b in B holds zz - b in X;
          thus contradiction by A14,A12,A13;
        end;
        hence x in (meet F)(-)B;
      end;
      hence (meet F)(-)B = meet {X(-)B where X is binary-image of E: X in F}
      by TARSKI:2;
    end;
  end;
