theorem Th58:
  LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is 0_Lattice
proof
  set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
  ex C being Element of S st for A being Element of S holds C "/\" A = C &
  A "/\" C = C
  proof
    reconsider C = (0).M as Element of S by Def3;
    take C;
    let A be Element of S;
    consider W being strict Subspace of M such that
A1: W = A by Def3;
    thus C "/\" A = SubMeet(M).(C,A) by LATTICES:def 2
      .= (0).M /\ W by A1,Def8
      .= C by Th20;
    thus A "/\" C = SubMeet(M).(A,C) by LATTICES:def 2
      .= W /\ (0).M by A1,Def8
      .= C by Th20;
  end;
  hence thesis by Th57,LATTICES:def 13;
end;
