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;

theorem Th29:
  (.p.> = <.p.:.) & (.p.:.> = <.p.)
proof
  (.p.> = .:<.p.:.)
  proof
    let q;
A1: q.: in <.p.:.) iff p.: [= q.: by FILTER_0:15;
    q in (.p.> iff q [= p by Th28;
    hence thesis by A1,LATTICE2:38,39;
  end;
  hence (.p.> = <.p.:.);
  .:(.p.:.> = <.p.)
  proof
    let q;
A2: q in <.p.) iff p [= q by FILTER_0:15;
    q.: in (.p.:.> iff q.: [= p.: by Th28;
    hence thesis by A2,LATTICE2:38,39;
  end;
  hence thesis;
end;
