
theorem LemM:
  for L being Lattice, I being Ideal of L holds
    I is max-ideal iff I is maximal
  proof
    let L be Lattice, I be Ideal of L;
    hereby
      assume
A0:   I is max-ideal; then
a2:   I <> the carrier of L by FILTER_2:def 8;
      for G being Ideal of L st G is proper & I c= G holds I = G
      proof
        let G be Ideal of L;
        assume
B1:     G is proper & I c= G; then
        G <> the carrier of L by SUBSET_1:def 6;
        hence thesis by FILTER_2:def 8,A0,B1;
      end;
      hence I is maximal by a2,SUBSET_1:def 6;
    end;
    assume
A0: I is maximal;
    for J being Ideal of L st I c= J & J <> the carrier of L holds I = J
    proof
      let J be Ideal of L;
      assume
B1:   I c= J & J <> the carrier of L; then
      J is proper by SUBSET_1:def 6;
      hence thesis by A0,B1;
    end;
    hence thesis by A0,FILTER_2:def 8,SUBSET_1:def 6;
  end;
