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;
