
theorem Th45:
  for J being set, D being non empty set, K being ManySortedSet of J
  for F being DoubleIndexedSet of K,D holds doms F = K
proof
  let J be set, D be non empty set, K be ManySortedSet of J;
  let F be DoubleIndexedSet of K,D;
A1: dom doms F = dom F by FUNCT_6:def 2;
A2: dom F = J by PARTFUN1:def 2;
A3: now
    let j be object;
    set f = F.j;
    assume
A4: j in J;
    then (J --> D).j = D by FUNCOP_1:7;
    then
A5: f is Function of K.j,D by A4,PBOOLE:def 15;
    (doms F).j = dom f by A2,A4,FUNCT_6:22;
    hence (doms F).j = K.j by A5,FUNCT_2:def 1;
  end;
  dom K = J & F"rng F = dom F by PARTFUN1:def 2,RELAT_1:134;
  hence thesis by A2,A1,A3,FUNCT_1:2;
end;
