let L be non empty transitive antisymmetric complete RelStr ; :: thesis: for S being non empty sups-closed Subset of L

for X being Subset of S holds "\/" (X,(subrelstr S)) = "\/" (X,L)

let S be non empty sups-closed Subset of L; :: thesis: for X being Subset of S holds "\/" (X,(subrelstr S)) = "\/" (X,L)

let X be Subset of S; :: thesis: "\/" (X,(subrelstr S)) = "\/" (X,L)

ex_sup_of X,L by YELLOW_0:17;

hence "\/" (X,(subrelstr S)) = "\/" (X,L) by Th22; :: thesis: verum

for X being Subset of S holds "\/" (X,(subrelstr S)) = "\/" (X,L)

let S be non empty sups-closed Subset of L; :: thesis: for X being Subset of S holds "\/" (X,(subrelstr S)) = "\/" (X,L)

let X be Subset of S; :: thesis: "\/" (X,(subrelstr S)) = "\/" (X,L)

ex_sup_of X,L by YELLOW_0:17;

hence "\/" (X,(subrelstr S)) = "\/" (X,L) by Th22; :: thesis: verum