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

theorem
  f is Functional_Sequence of D1, D2
  iff (dom f = NAT & for x st x in NAT holds f.x is PartFunc of D1, D2)
  proof
    thus f is Functional_Sequence of D1, D2
    implies (dom f = NAT & for x st x in NAT holds f.x is PartFunc of D1, D2)
    proof
      assume
      A1: f is Functional_Sequence of D1, D2;
      hence dom f = NAT by FUNCT_2:def 1;
      let x;
      assume x in NAT;
      then x in dom f by A1,FUNCT_2:def 1; then
      A2: f.x in rng f by FUNCT_1:def 3;
      rng f c= PFuncs(D1, D2) by A1,RELAT_1:def 19;
      hence thesis by A2, PARTFUN1:46;
    end;
    assume that
    A3: dom f = NAT and
    A4: for x st x in NAT holds f.x is PartFunc of D1, D2;
    now
      let y be object;
      assume y in rng f;
      then consider x being object such that
      A5: x in dom f and
      A6: y = f.x by FUNCT_1:def 3;
      f.x is PartFunc of D1, D2 by A3, A4, A5;
      hence y in PFuncs(D1, D2) by A6, PARTFUN1:45;
    end;
    then rng f c=PFuncs(D1, D2);
    hence thesis by A3,FUNCT_2:def 1,RELSET_1:4;
  end;
