reserve a, b for Real;
reserve RNS for RealNormSpace;
reserve x, y, z, g, g1, g2 for Point of RNS;
reserve S, S1, S2 for sequence of RNS;
reserve k, n, m, m1, m2 for Nat;
reserve r for Real;
reserve f for Function;
reserve d, s, t for set;

theorem
  for RNS being non empty 1-sorted, x being Element of RNS holds f is
  sequence of RNS iff ( dom f = NAT & for d st d in NAT holds f.d is Element of
  RNS )
proof
  let RNS be non empty 1-sorted;
  let x be Element of RNS;
  thus f is sequence of RNS implies ( dom f = NAT & for d st d in NAT holds f.
  d is Element of RNS )
  proof
    assume
A1: f is sequence of RNS;
    then
A2: rng f c= the carrier of RNS by RELAT_1:def 19;
A3: dom f = NAT by A1,FUNCT_2:def 1;
    for d st d in NAT holds f.d is Element of RNS
    proof
      let d;
      assume d in NAT;
      then f.d in rng f by A3,FUNCT_1:def 3;
      hence thesis by A2;
    end;
    hence thesis by A1,FUNCT_2:def 1;
  end;
  thus ( dom f = NAT & for d st d in NAT holds f.d is Element of RNS ) implies
  f is sequence of RNS
  proof
    assume that
A4: dom f = NAT and
A5: for d st d in NAT holds f.d is Element of RNS;
    for s being object st s in rng f holds s in the carrier of RNS
    proof
      let s be object;
      assume s in rng f;
      then consider d being object such that
A6:   d in dom f and
A7:   s = f.d by FUNCT_1:def 3;
      f.d is Element of RNS by A4,A5,A6;
      hence thesis by A7;
    end;
    then rng f c= the carrier of RNS by TARSKI:def 3;
    hence thesis by A4,FUNCT_2:def 1,RELSET_1:4;
  end;
end;
