
theorem
  for T being complete LATTICE
  for S being sups-inheriting full non empty SubRelStr of T
  holds incl(S,T) is sups-preserving
proof
  let T be complete LATTICE;
  let S be sups-inheriting full non empty SubRelStr of T;
  set f = incl(S,T);
  let X be Subset of S;
  assume ex_sup_of X, S;
  thus ex_sup_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_sup_of X, T by YELLOW_0:17;
A4: f.sup X = sup X by A1;
  "\/"(X,T) in the carrier of S by A3,YELLOW_0:def 19;
  hence thesis by A2,A3,A4,YELLOW_0:64;
end;
