reserve R for Ring,
  V for RightMod of R,
  W,W1,W2,W3 for Submodule of V,
  u,u1, u2,v,v1,v2 for Vector of V,
  x,y,y1,y2 for object;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
reserve A1,A2,B for Element of Submodules(V);

theorem Th48:
  LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #) is 0_Lattice
proof
  set S = LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #);
  ex C being Element of S st for A being Element of S holds C "/\" A = C &
  A "/\" C = C
  proof
    reconsider C = (0).V as Element of S by Def3;
    take C;
    let A be Element of S;
    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;
    thus A "/\" C = SubMeet(V).(A,C) by LATTICES:def 2
      .= W /\ (0).V by A1,Def7
      .= C by Th21;
  end;
  hence thesis by Th47,LATTICES:def 13;
end;
