let L be non empty transitive antisymmetric complete RelStr ; :: thesis: for S being non empty full sups-inheriting SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X = finsups Y
let S be non empty full sups-inheriting SubRelStr of L; :: thesis: for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X = finsups Y
let X be Subset of L; :: thesis: for Y being Subset of S st X = Y holds
finsups X = finsups Y
let Y be Subset of S; :: thesis: ( X = Y implies finsups X = finsups Y )
assume A1: X = Y ; :: thesis: finsups X = finsups Y
hence finsups X c= finsups Y by Th3; :: according to XBOOLE_0:def 10 :: thesis: finsups Y c= finsups X
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in finsups Y or x in finsups X )
assume x in finsups Y ; :: thesis: x in finsups X
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 being finite Subset of Y such that
A2: x = "\/" (Z,S) and
ex_sup_of Z,S ;
reconsider Z = 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 x in finsups X by WAYBEL_0:def 27; :: thesis: verum