
theorem Th40: :: Theorem 2.11, p. 61-62 (3) implies (1)
:: Not in CCL, consequence of other implications.
  for L being continuous complete LATTICE, k being
  directed-sups-preserving kernel Function of L, L holds kernel_congruence k is
  CLCongruence
proof
  let L be continuous complete LATTICE, k be directed-sups-preserving kernel
  Function of L, L;
  set R = kernel_congruence k;
  thus
A1: R is Equivalence_Relation of the carrier of L by Th2;
  ex LR being continuous complete strict LATTICE st the carrier of LR =
Class EqRel R & the InternalRel of LR = {[Class(EqRel R, x), Class(EqRel R, y)]
  where x, y is Element of L : k.x <= k.y } & for g being Function of L, LR st
  for x being Element of L holds g.x = Class(EqRel R, x) holds g is
  CLHomomorphism of L, LR by Th37;
  hence thesis by A1,Th38;
end;
