
theorem Th3:
  for L be non empty transitive 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 c= finsups Y
proof
  let L be non empty transitive 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;
  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;
  reconsider Z1 = Z as Subset of S by XBOOLE_1:1;
A4: "\/"(Z1,L) in the carrier of S by A3,YELLOW_0:def 19;
  then
A5: ex_sup_of Z1,S by A3,YELLOW_0:64;
  x = "\/"(Z1,S) by A2,A3,A4,YELLOW_0:64;
  then x in {"\/"(V,S) where V is finite Subset of Y: ex_sup_of V,S} by A5;
  hence thesis by WAYBEL_0:def 27;
end;
