thus id L is sups-preserving :: thesis: id L is infs-preserving

proof

let X be Subset of L; :: according to WAYBEL_0:def 33 :: thesis: id L preserves_sup_of X
A1: (id L) . (sup X) = sup X by FUNCT_1:35;
assume ex_sup_of X,L ; :: according to WAYBEL_0:def 31 :: thesis: ( ex_sup_of (id L) .: X,L & "\/" ((id L) .: X),L = (id L) . ("\/" X,L) )
hence ( ex_sup_of (id L) .: X,L & "\/" ((id L) .: X),L = (id L) . ("\/" X,L) ) by A1, FUNCT_1:162; :: thesis: verum

end;

let X be Subset of L; :: according to WAYBEL_0:def 32 :: thesis: id L preserves_inf_of X
A2: (id L) . (inf X) = inf X by FUNCT_1:35;
assume ex_inf_of X,L ; :: according to WAYBEL_0:def 30 :: thesis: ( ex_inf_of (id L) .: X,L & "/\" ((id L) .: X),L = (id L) . ("/\" X,L) )
hence ( ex_inf_of (id L) .: X,L & "/\" ((id L) .: X),L = (id L) . ("/\" X,L) ) by A2, FUNCT_1:162; :: thesis: verum