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;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem Th62:
  p [= q implies (r in [#p,q#] iff p [= r & r [= q)
proof
  assume p [= q;
  then [#p,q#] = {s where s is Element of L: p [= s & s [= q} by Def12;
  then r in [#p,q#] iff ex s being Element of L st r = s & p [= s & s [= q;
  hence thesis;
end;
