
theorem
  for L be complete transitive antisymmetric non empty RelStr for S be
  sups-inheriting non empty full SubRelStr of L for X be Subset of L for Y be
  Subset of S st X = Y holds finsups X = finsups Y
proof
  let L be complete transitive antisymmetric non empty RelStr;
  let S be sups-inheriting non empty full SubRelStr of L;
  let X be Subset of L;
  let Y be Subset of S;
  assume
A1: X = Y;
  hence finsups X c= finsups Y by Th3;
  let x be object;
  assume x in finsups Y;
  then x in {"\/"(V,S) where V is finite Subset of Y: ex_sup_of V,S} by
WAYBEL_0:def 27;
  then consider Z be finite Subset of Y such that
A2: x = "\/"(Z,S) and
  ex_sup_of Z,S;
  reconsider Z as finite Subset of X by A1;
  reconsider Z1 = Z as Subset of S by XBOOLE_1:1;
A3: ex_sup_of Z1,L by YELLOW_0:17;
  then "\/"(Z1,L) in the carrier of S by YELLOW_0:def 19;
  then x = "\/"(Z1,L) by A2,A3,YELLOW_0:64;
  then x in {"\/"(V,L) where V is finite Subset of X: ex_sup_of V,L} by A3;
  hence thesis by WAYBEL_0:def 27;
end;
