reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th2:
  for X,Y,D be set, F be Functional_Sequence of X,Y st F is
  with_the_same_dom holds F||D is with_the_same_dom
proof
  let X,Y,D be set, F be Functional_Sequence of X,Y;
  assume
A1: F is with_the_same_dom;
  let n,m be Nat;
  set G = F||D;
  G.m = (F.m)|D by Def1;
  then
A2: dom(G.m) = dom(F.m) /\ D by RELAT_1:61;
  G.n = (F.n)|D by Def1;
  then dom(G.n) = dom(F.n) /\ D by RELAT_1:61;
  hence dom(G.n) = dom(G.m) by A1,A2;
end;
