reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;

theorem
  f is Functional_Sequence of D1,D2 iff dom f = NAT & for n holds f.n
  is PartFunc of D1,D2
proof
  thus f is Functional_Sequence of D1,D2 implies dom f=NAT & for n holds f.n
  is PartFunc of D1,D2
  by Lm1,ORDINAL1:def 12;
  assume that
A1: dom f=NAT and
A2: for n holds f.n is PartFunc of D1,D2;
  for x holds x in NAT implies f.x is PartFunc of D1,D2 by A2;
  hence thesis by A1,Lm1;
end;
