
theorem Th26:
  for S,T being complete LATTICE holds UPS(S, T) is
  sups-inheriting SubRelStr of T|^the carrier of S
proof
  let S,T be complete LATTICE;
  set S9 = the Scott TopAugmentation of S;
  set T9 = the Scott TopAugmentation of T;
A1: the RelStr of T = the RelStr of T9 by YELLOW_9:def 4;
  then
A2: T9|^the carrier of S = T|^the carrier of S by Th15;
A3: the RelStr of S = the RelStr of S9 by YELLOW_9:def 4;
  then UPS(S,T) = UPS(S9,T9) by A1,Th25
    .= SCMaps(S9,T9) by Th24
    .= ContMaps(S9,T9) by WAYBEL24:38;
  hence thesis by A2,A3,WAYBEL24:35;
end;
