 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th26:
for X being NormedLinearTopSpace,
    RNS being RealNormSpace,
    t being sequence of X,
    s being sequence of RNS
st RNS = the NORMSTR of X & t=s
   & t is convergent
 holds s is convergent & lim s = lim t
proof
let X be NormedLinearTopSpace,
    RNS be RealNormSpace,
    t2 be sequence of X,
    s2 be sequence of RNS;
assume A1: RNS = the NORMSTR of X
           & t2=s2
           & t2 is convergent;
         reconsider y = lim t2 as Point of RNS by A1;
      A2:for r being Real st 0 < r holds
         ex m being Nat st
         for n being Nat st m <= n holds
         ||.s2.n - y .|| < r
      proof
         let r be Real;
         assume 0 < r; then
         consider m being Nat such that
         A3:   for n being Nat st m <= n holds
                ||.t2 . n - lim t2 .|| < r by A1,Th22;
        take m;
        let n be Nat;
        assume m <= n; then
        ||.t2 . n - lim t2 .|| < r by A3;
        hence ||.s2.n - y .|| < r by Th19,A1;
      end;
      hence s2 is convergent;
      hence thesis by A2,NORMSP_1:def 7;
end;
