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;
