let T be complete LATTICE; :: thesis: for S being non empty full filtered-infs-inheriting SubRelStr of T holds incl (S,T) is filtered-infs-preserving
let S be non empty full filtered-infs-inheriting SubRelStr of T; :: thesis: incl (S,T) is filtered-infs-preserving
set f = incl (S,T);
let X be Subset of S; :: according to WAYBEL_0:def 36 :: thesis: ( X is empty or not X is filtered or incl (S,T) preserves_inf_of X )
assume that
A1: ( not X is empty & X is filtered ) and
ex_inf_of X,S ; :: according to WAYBEL_0:def 30 :: thesis: ( ex_inf_of (incl (S,T)) .: X,T & "/\" (((incl (S,T)) .: X),T) = (incl (S,T)) . ("/\" (X,S)) )
thus ex_inf_of (incl (S,T)) .: X,T by YELLOW_0:17; :: thesis: "/\" (((incl (S,T)) .: X),T) = (incl (S,T)) . ("/\" (X,S))
the carrier of S c= the carrier of T by YELLOW_0:def 13;
then A2: incl (S,T) = id the carrier of S by YELLOW_9:def 1;
then A3: (incl (S,T)) .: X = X by FUNCT_1:92;
A4: ex_inf_of X,T by YELLOW_0:17;
(incl (S,T)) . (inf X) = inf X by A2;
hence "/\" (((incl (S,T)) .: X),T) = (incl (S,T)) . ("/\" (X,S)) by A1, A3, A4, WAYBEL_0:6; :: thesis: verum