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 Th61:
  o1 = (the L_join of L)||P & o2 = (the L_meet of L)||P implies o1
is commutative & o1 is associative & o2 is commutative & o2 is associative & o1
  absorbs o2 & o2 absorbs o1 by Th1,Th5,LATTICE2:26,27;
