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, S being sequence of RNS st (ex x
  being Element of RNS st for n being Nat holds S.n = x) ex x being Element of
  RNS st rng S={x}
proof
  let RNS be non empty 1-sorted;
  let S be sequence of RNS;
  given z being Element of RNS such that
A1: for n being Nat holds S.n = z;
  take x = z;
  now
    let s be object;
    assume s in {x}; then
A2: s = x by TARSKI:def 1;
    now
      assume
A3:   not s in rng S;
      now
        let n;
        n in NAT by ORDINAL1:def 12;
        then n in dom S by FUNCT_2:def 1;
        then S.n <> x by A2,A3,FUNCT_1:def 3;
        hence contradiction by A1;
      end;
      hence contradiction;
    end;
    hence s in rng S;
  end;
  then
A4: {x} c= rng S by TARSKI:def 3;
  now
    let t be object;
    assume t in rng S;
    then consider d being object such that
A5: d in dom S and
A6: S.d = t by FUNCT_1:def 3;
    d in NAT by A5,FUNCT_2:def 1;
    then t = z by A1,A6;
    hence t in {x} by TARSKI:def 1;
  end;
  then rng S c= {x} by TARSKI:def 3;
  hence thesis by A4,XBOOLE_0:def 10;
end;
