theorem Th66:
  L is I_Lattice implies equivalence_wrt F is_transitive_in the carrier of L
proof
  assume
A1: L is I_Lattice;
  let x,y,z be object;
  assume x in the carrier of L & y in the carrier of L & z in the carrier of L;
  then reconsider p = x, q = y, r = z as Element of L;
  assume [x,y] in equivalence_wrt F & [y,z] in equivalence_wrt F;
  then p <=> q in F & q <=> r in F by Def11;
  then p <=> r in F by A1,Th62;
  hence thesis by Def11;
end;
