reserve x, y, i for object,
  L for up-complete Semilattice;

theorem Th6:
  for J, D being set, K being ManySortedSet of J for F being
DoubleIndexedSet of K, D for j being set st j in J holds F.j is Function of K.j
  , D
proof
  let J, D be set, K be ManySortedSet of J;
  let F be DoubleIndexedSet of K, D;
  let j be set;
  assume
A1: j in J;
  then (J --> D).j = D by FUNCOP_1:7;
  hence thesis by A1,PBOOLE:def 15;
end;
