reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;

theorem Th52:
  L is C_Lattice iff L.: is C_Lattice
proof
A1: L is upper-bounded iff L.: is lower-bounded by LATTICE2:49;
A2: L is lower-bounded iff L.: is upper-bounded by LATTICE2:48;
  thus L is C_Lattice implies L.: is C_Lattice
  proof
    assume
A3: L is C_Lattice;
    now
      let p9;
      consider q such that
A4:   q is_a_complement_of .:p9 by A3,LATTICES:def 19;
      take r = q.:;
      q"\/".: p9 = Top L by A4;
      then
A5:   q.:"/\"p9 = Top L;
      q"/\".:p9 = Bottom L by A4;
      then
A6:   q.:"\/"p9 = Bottom L;
      Top (L.: ) = Bottom L & Bottom (L.:) = Top L by A3,LATTICE2:61,62;
      hence r is_a_complement_of p9 by A5,A6;
    end;
    hence thesis by A2,A1,A3,LATTICES:def 19;
  end;
  assume
A7: L.: is C_Lattice;
  now
    let p;
    consider q9 such that
A8: q9 is_a_complement_of p.: by A7,LATTICES:def 19;
    q9"\/"p.: = Top (L.:) by A8;
    then
A9: .:q9"/\"p = Top (L.:);
    take r = .:q9;
    L is upper-bounded by A7,LATTICE2:49;
    then
A10: Bottom (L.:) = Top L by LATTICE2:62;
    q9"/\"p.: = Bottom (L.:) by A8;
    then
A11: .:q9"\/"p = Bottom (L.:);
    L is lower-bounded by A7,LATTICE2:48;
    then Top (L.: ) = Bottom L by LATTICE2:61;
    hence r is_a_complement_of p by A9,A11,A10;
  end;
  hence thesis by A2,A1,A7,LATTICES:def 19;
end;
