
theorem Th65:
  for S,T being lower-bounded LATTICE
  for f being finite-sups-preserving Function of S,T
  for S9 being non empty finite-sups-inheriting full SubRelStr of S
  for T9 being non empty finite-sups-inheriting full SubRelStr of T
  for g being Function of S9, T9 st g = f|the carrier of S9
  holds g is finite-sups-preserving
proof
  let S,T be lower-bounded LATTICE;
  let f be finite-sups-preserving Function of S,T;
  let S9 be non empty finite-sups-inheriting full SubRelStr of S;
  let T9 be non empty finite-sups-inheriting full SubRelStr of T;
  let g be Function of S9, T9 such that
A1: g = f|the carrier of S9;
  Bottom S9 = Bottom S by Th60;
  then g.Bottom S9 = f.Bottom S by A1,FUNCT_1:49
    .= Bottom T by Def17
    .=
  Bottom T9 by Th60;
  then g is bottom-preserving;
  hence thesis by A1,Th62,Th64;
end;
