let L be non empty transitive RelStr ; :: thesis: for S being non empty full filtered-infs-inheriting SubRelStr of L

for X being filtered Subset of S st X <> {} & ex_inf_of X,L holds

( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

let S be non empty full filtered-infs-inheriting SubRelStr of L; :: thesis: for X being filtered Subset of S st X <> {} & ex_inf_of X,L holds

( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

let X be filtered Subset of S; :: thesis: ( X <> {} & ex_inf_of X,L implies ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) ) )

assume that

A1: X <> {} and

A2: ex_inf_of X,L ; :: thesis: ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

"/\" (X,L) in the carrier of S by A1, A2, Def3;

hence ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) ) by A2, YELLOW_0:63; :: thesis: verum

for X being filtered Subset of S st X <> {} & ex_inf_of X,L holds

( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

let S be non empty full filtered-infs-inheriting SubRelStr of L; :: thesis: for X being filtered Subset of S st X <> {} & ex_inf_of X,L holds

( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

let X be filtered Subset of S; :: thesis: ( X <> {} & ex_inf_of X,L implies ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) ) )

assume that

A1: X <> {} and

A2: ex_inf_of X,L ; :: thesis: ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) )

"/\" (X,L) in the carrier of S by A1, A2, Def3;

hence ( ex_inf_of X,S & "/\" (X,S) = "/\" (X,L) ) by A2, YELLOW_0:63; :: thesis: verum