reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th62:
  i <=> j in FI & j <=> k in FI implies i <=> k in FI
proof
  assume
A1: i <=> j in FI & j <=> k in FI;
  then j => i in FI & k => j in FI by Th8;
  then
A2: k => i in FI by Th41;
  i => j in FI & j => k in FI by A1,Th8;
  then i => k in FI by Th41;
  hence thesis by A2,Th8;
end;
