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 Th49:
  for V being RightMod of R holds LattStr (# Submodules(V),
    SubJoin(V), SubMeet(V) #) is 1_Lattice
proof
  let V be RightMod of R;
  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
    consider W9 being strict Submodule of V such that
A1: the carrier of W9 = the carrier of (Omega).V;
    reconsider C = W9 as Element of S by Def3;
    take C;
    let A be Element of S;
    consider W being strict Submodule of V such that
A2: W = A by Def3;
A3: C is Submodule of (Omega).V by Lm5;
    thus C "\/" A = SubJoin(V).(C,A) by LATTICES:def 1
      .= W9 + W by A2,Def6
      .= (Omega).V + W by A1,Lm4
      .= the RightModStr of V by Th11
      .= C by A1,A3,RMOD_2:31;
    thus A "\/" C = SubJoin(V).(A,C) by LATTICES:def 1
      .= W + W9 by A2,Def6
      .= W + (Omega).V by A1,Lm4
      .= the RightModStr of V by Th11
      .= C by A1,A3,RMOD_2:31;
  end;
  hence thesis by Th47,LATTICES:def 14;
end;
