let L be non empty RelStr ; :: thesis: for X being set st ( ex_sup_of X,L or ex_sup_of X /\ the carrier of L,L ) holds
"\/" X,L = "\/" (X /\ the carrier of L),L
let X be set ; :: thesis: ( ( ex_sup_of X,L or ex_sup_of X /\ the carrier of L,L ) implies "\/" X,L = "\/" (X /\ the carrier of L),L )
set Y = X /\ the carrier of L;
assume ( ex_sup_of X,L or ex_sup_of X /\ the carrier of L,L ) ; :: thesis: "\/" X,L = "\/" (X /\ the carrier of L),L
then ( ex_sup_of X,L & ( for x being Element of L holds
( x is_>=_than X iff x is_>=_than X /\ the carrier of L ) ) ) by Th5, Th50;
hence "\/" X,L = "\/" (X /\ the carrier of L),L by Th47; :: thesis: verum