let T be non empty TopSpace; :: thesis: for F being Subset-Family of T st F is closed-domains-family holds
for X being Subset of (Domains_Lattice T) st X = F holds
( ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) & ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) )
let F be Subset-Family of T; :: thesis: ( F is closed-domains-family implies for X being Subset of (Domains_Lattice T) st X = F holds
( ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) & ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) ) )
assume A1: F is closed-domains-family ; :: thesis: for X being Subset of (Domains_Lattice T) st X = F holds
( ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) & ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) )
then A2: F is domains-family by Th73;
A3: meet F is closed by A1, Th74, TOPS_2:29;
Cl (Int (meet F)) c= Cl (meet F) by PRE_TOPC:49, TOPS_1:44;
then Cl (Int (meet F)) c= meet F by A3, PRE_TOPC:52;
then A4: (meet F) /\ (Cl (Int (meet F))) = Cl (Int (meet F)) by XBOOLE_1:28;
let X be Subset of (Domains_Lattice T); :: thesis: ( X = F implies ( ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) & ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) ) )
assume A5: X = F ; :: thesis: ( ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) & ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) )
hence ( X <> {} implies "/\" X,(Domains_Lattice T) = Cl (Int (meet F)) ) by A2, A4, Th93; :: thesis: ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T )
thus ( X = {} implies "/\" X,(Domains_Lattice T) = [#] T ) by A2, A5, Th93; :: thesis: verum