reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th6:
  for seq be sequence of S, x be set holds x in rng seq iff ex n st x = seq.n
proof
  let seq be sequence of S;
  let x be set;
  thus x in rng seq implies ex n 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 Nat by A1;
    take m;
    thus thesis by A2;
  end;
  given n 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;
