
theorem
  for T being complete LATTICE
  for S being filtered-infs-inheriting full non empty SubRelStr of T
  holds incl(S,T) is filtered-infs-preserving
proof
  let T be complete LATTICE;
  let S be filtered-infs-inheriting full non empty SubRelStr of T;
  set f = incl(S,T);
  let X be Subset of S;
  assume that
A1: X is non empty filtered and ex_inf_of X, S;
  thus ex_inf_of f.:X, T by YELLOW_0:17;
  the carrier of S c= the carrier of T by YELLOW_0:def 13;
  then
A2: f = id the carrier of S by YELLOW_9:def 1;
  then
A3: f.:X = X by FUNCT_1:92;
A4: ex_inf_of X, T by YELLOW_0:17;
  f.inf X = inf X by A2;
  hence thesis by A1,A3,A4,WAYBEL_0:6;
end;
