
theorem Th16:
  for T being upper-bounded Semilattice
  for S being meet-inheriting full non empty SubRelStr of T
  st Top T in the carrier of S & S is filtered-infs-inheriting
  holds S is infs-inheriting
proof
  let T be upper-bounded Semilattice;
  let S be meet-inheriting full non empty SubRelStr of T such that
A1: Top T in the carrier of S and
A2: S is filtered-infs-inheriting;
  let A be Subset of S;
  the carrier of S c= the carrier of T by YELLOW_0:def 13;
  then reconsider C = A as Subset of T by XBOOLE_1:1;
  set F = fininfs C;
  assume
A3: ex_inf_of A, T;
  then
A4: inf F = inf C by WAYBEL_0:60;
  F c= the carrier of S
  proof
    let x be object;
    assume x in F;
    then consider Y being finite Subset of C such that
A5: x = "/\"(Y, T) and ex_inf_of Y, T;
    reconsider Y as finite Subset of T by XBOOLE_1:1;
    reconsider Z = Y as finite Subset of S by XBOOLE_1:1;
    assume
A6: not x in the carrier of S;
    then Z <> {} by A1,A5;
    hence thesis by A5,A6,Th14;
  end;
  then reconsider G = F as non empty Subset of S;
  reconsider G as filtered non empty Subset of S by WAYBEL10:23;
A7: now
    let Y be finite Subset of C;
    Y c= the carrier of T by XBOOLE_1:1;
    hence Y <> {} implies ex_inf_of Y,T by YELLOW_0:55;
  end;
A8: now
    let x be Element of T;
    assume x in F;
    then ex Y being finite Subset of C st x = "/\"(Y,T) & ex_inf_of Y,T;
    hence ex Y being finite Subset of C st ex_inf_of Y,T & x = "/\"(Y,T);
  end;
  now
    let Y be finite Subset of C;
    reconsider Z = Y as finite Subset of T by XBOOLE_1:1;
    assume Y <> {};
    then ex_inf_of Z, T by YELLOW_0:55;
    hence "/\"(Y,T) in F;
  end;
  then ex_inf_of G, T by A3,A7,A8,WAYBEL_0:58;
  hence thesis by A2,A4;
end;
