let R be Ring; :: thesis: for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 0_Lattice
let V be RightMod of R; :: thesis: LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 0_Lattice
set S = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #);
ex C being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) st
for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "/\" A = C & A "/\" C = C )

proof

reconsider C = (0). V as Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) by Def3;
take C ; :: thesis: for A being Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) holds
( C "/\" A = C & A "/\" C = C )
let A be Element of LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #); :: thesis: ( C "/\" A = C & A "/\" C = C )
consider W being strict Submodule of V such that
A1: W = A by Def3;
thus C "/\" A = (SubMeet V) . (C,A) by LATTICES:def 2
.= ((0). V) /\ W by A1, Def7
.= C by Th21 ; :: thesis: A "/\" C = C
thus A "/\" C = (SubMeet V) . (A,C) by LATTICES:def 2
.= W /\ ((0). V) by A1, Def7
.= C by Th21 ; :: thesis: verum

end;

hence LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 0_Lattice by Th47, LATTICES:def 13; :: thesis: verum