set t = Bottom T;
set f = S --> (Bottom T);
take S --> (Bottom T) ; :: thesis: S --> (Bottom T) is sups-preserving
let X be Subset of S; :: according to WAYBEL_0:def 33 :: thesis: S --> (Bottom T) preserves_sup_of X
assume ex_sup_of X,S ; :: according to WAYBEL_0:def 31 :: thesis: ( ex_sup_of (S --> (Bottom T)) .: X,T & "\/" (((S --> (Bottom T)) .: X),T) = (S --> (Bottom T)) . ("\/" (X,S)) )
thus ex_sup_of (S --> (Bottom T)) .: X,T by YELLOW_0:17; :: thesis: "\/" (((S --> (Bottom T)) .: X),T) = (S --> (Bottom T)) . ("\/" (X,S))
( (S --> (Bottom T)) .: X c= rng (S --> (Bottom T)) & rng (S --> (Bottom T)) c= {(Bottom T)} ) by FUNCOP_1:13, RELAT_1:111;
then (S --> (Bottom T)) .: X c= {(Bottom T)} ;
then A1: ( (S --> (Bottom T)) .: X = {(Bottom T)} or (S --> (Bottom T)) .: X = {} ) by ZFMISC_1:33;
per cases ( X = {} or X <> {} ) ;
suppose X = {} ; :: thesis: "\/" (((S --> (Bottom T)) .: X),T) = (S --> (Bottom T)) . ("\/" (X,S))
then (S --> (Bottom T)) .: X = {} ;
then sup ((S --> (Bottom T)) .: X) = Bottom T by YELLOW_0:def 11;
hence "\/" (((S --> (Bottom T)) .: X),T) = (S --> (Bottom T)) . ("\/" (X,S)) ; :: thesis: verum
end;
suppose X <> {} ; :: thesis: "\/" (((S --> (Bottom T)) .: X),T) = (S --> (Bottom T)) . ("\/" (X,S))
then reconsider X1 = X as non empty Subset of S ;
set x = the Element of X1;
(S --> (Bottom T)) . the Element of X1 in (S --> (Bottom T)) .: X by FUNCT_2:35;
hence sup ((S --> (Bottom T)) .: X) = Bottom T by A1, YELLOW_0:39
.= (S --> (Bottom T)) . (sup X) ;
:: thesis: verum
end;
end;