
theorem Th8:
  for T being complete LATTICE
  for S being infs-inheriting full non empty SubRelStr of T
  holds incl(S,T) is infs-preserving
proof
  let T be complete LATTICE;
  let S be infs-inheriting full non empty SubRelStr of T;
  set f = incl(S,T);
  let X be Subset of S;
  assume 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
A1: f = id the carrier of S by YELLOW_9:def 1;
  then
A2: f.:X = X by FUNCT_1:92;
A3: ex_inf_of X, T by YELLOW_0:17;
A4: f.inf X = inf X by A1;
  "/\"(X,T) in the carrier of S by A3,YELLOW_0:def 18;
  hence thesis by A2,A3,A4,YELLOW_0:63;
end;
