reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;
reserve B for B_Lattice,
  a,b,c,d for Element of B;

theorem Th59:
  i in Class(equivalence_wrt FI,k) & j in Class(equivalence_wrt FI
  ,k) implies i"\/"j in Class(equivalence_wrt FI,k) & i"/\" j in Class(
  equivalence_wrt FI,k)
proof
  assume that
A1: i in Class(equivalence_wrt FI,k) and
A2: j in Class(equivalence_wrt FI,k);
A3: i <=> k in FI by A1,Lm4;
A4: j <=> k in FI by A2,Lm4;
  k"/\"k = k;
  then
A5: (i"/\"j) <=> k in FI by A3,A4,Th58;
  k"\/"k = k;
  then (i"\/"j) <=> k in FI by A3,A4,Th58;
  hence thesis by A5,Lm4;
end;
