
theorem Th2:
  for L be non empty RelStr for X,Y be Subset of L st X c= Y holds
  finsups X c= finsups Y
proof
  let L be non empty RelStr;
  let X,Y be Subset of L;
  assume
A1: X c= Y;
  let x be object;
  assume x in finsups X;
  then x in {"\/"(V,L) where V is finite Subset of X: ex_sup_of V,L} by
WAYBEL_0:def 27;
  then consider Z be finite Subset of X such that
A2: x = "\/"(Z,L) and
A3: ex_sup_of Z,L;
  reconsider Z as finite Subset of Y by A1,XBOOLE_1:1;
  ex_sup_of Z,L by A3;
  then x in {"\/"(V,L) where V is finite Subset of Y: ex_sup_of V,L} by A2;
  hence thesis by WAYBEL_0:def 27;
end;
