let S be non empty RelStr ; :: thesis: for T being non empty reflexive antisymmetric upper-bounded RelStr holds S --> (Top T) is infs-preserving
let T be non empty reflexive antisymmetric upper-bounded RelStr ; :: thesis: S --> (Top T) is infs-preserving
let X be Subset of S; :: according to WAYBEL_0:def 32 :: thesis: S --> (Top T) preserves_inf_of X
assume ex_inf_of X,S ; :: according to WAYBEL_0:def 30 :: thesis: ( ex_inf_of (S --> (Top T)) .: X,T & "/\" (((S --> (Top T)) .: X),T) = (S --> (Top T)) . ("/\" (X,S)) )
set t = Top T;
set f = the carrier of S --> (Top T);
A1: ( the carrier of S --> (Top T)) . (inf X) = Top T by FUNCOP_1:7;
(S --> (Top T)) .: X c= {(Top T)} by FUNCOP_1:81;
then ( (S --> (Top T)) .: X = {(Top T)} or (S --> (Top T)) .: X = {} ) by ZFMISC_1:33;
hence ( ex_inf_of (S --> (Top T)) .: X,T & "/\" (((S --> (Top T)) .: X),T) = (S --> (Top T)) . ("/\" (X,S)) ) by A1, YELLOW_0:38, YELLOW_0:39, YELLOW_0:43; :: thesis: verum