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
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 FUNCT_1:92; :: thesis: verum

end;

let X be Subset of L; :: according to WAYBEL_0:def 32 :: thesis: id L preserves_inf_of X
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 FUNCT_1:92; :: thesis: verum