reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;

theorem Th7:
  for seq be sequence of CNS, x be set holds x in rng seq
  iff ex n being Nat st x = seq.n
proof
  let seq be sequence of CNS;
  let x be set;
  thus x in rng seq implies ex n being Nat st x = seq.n
  proof
    assume x in rng seq;
    then consider y be object such that
A1: y in dom seq and
A2: x = seq.y by FUNCT_1:def 3;
    reconsider m=y as Element of NAT by A1;
    take m;
    thus thesis by A2;
  end;
  given n being Nat such that
A3: x = seq.n;
  n in NAT by ORDINAL1:def 12;
  then n in dom seq by FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:def 3;
end;
