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 Th67:
  L is I_Lattice implies equivalence_wrt F is Equivalence_Relation
  of the carrier of L
proof
  reconsider R = equivalence_wrt F as Relation of the carrier of L by Th63;
A1: R is_symmetric_in the carrier of L by Th65;
  assume
A2: L is I_Lattice;
  then R is_reflexive_in the carrier of L by Th64;
  then
A3: field R = the carrier of L & dom R = the carrier of L by ORDERS_1:13;
  R is_transitive_in the carrier of L by A2,Th66;
  hence thesis by A3,A1,PARTFUN1:def 2,RELAT_2:def 11,def 16;
end;
