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 Th14:
  (meet F)(+)B c= 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;
      (meet F) = {} by A1,SETFAM_1:def 1;
      then Z(+)B ={} by Th1;
      hence (meet F)(+)B
      c= meet {X(+)B where X is binary-image of E: X in F};
    end;
    suppose
      F <> {}; then
      consider W0 be object such that
      A2: W0 in F by XBOOLE_0:def 1;
      reconsider W0 as binary-image of E by A2;
      A3: W0(+)B in {W(+)B where W is binary-image of E: W in F} by A2;
       let x be object;
        assume x in (meet F)(+)B;
        then consider f, b be Element of E such that
        A4: x = f + b & f in (meet F) & b in B;
        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
          A5: Y = X(+)B & X in F;
          meet F c= X by A5,SETFAM_1:3;
          hence x in Y by A4,A5;
        end;
        hence x in meet {W(+)B where W is binary-image of E: W in F}
        by A3,SETFAM_1:def 1;
    end;
  end;
