
theorem Th3:
  for A being set, F being Subset-Family of A, R be Relation holds
    R.: meet F c= meet {R.:X where X is Subset of A : X in F}
proof
  let A be set, F be Subset-Family of A, R be Relation;
  per cases;
  suppose meet F = {};
    then R.: meet F = {};
    hence thesis by XBOOLE_1:2;
  end;
  suppose meet F <> {};
    then consider X0 being object such that
      A1: X0 in F by SETFAM_1:1, XBOOLE_0:def 1;
    reconsider X0 as Subset of A by A1;
    A2: R.:X0 in {R.:X where X is Subset of A : X in F} by A1;
      let y be object;
      assume y in R.: meet F;
      then consider x being object such that
        A3: [x,y] in R & x in meet F by RELAT_1:def 13;
      now
        let Y be set;
        assume Y in {R.:X where X is Subset of A : X in F};
        then consider X being Subset of A such that
          A4: Y = R.:X & X in F;
        x in X by A3, A4, SETFAM_1:def 1;
        hence y in Y by A3, A4, RELAT_1:def 13;
      end;
      hence y in meet {R.:X where X is Subset of A : X in F}
        by A2, SETFAM_1:def 1;
  end;
end;
