reserve E,F,G for RealNormSpace;
reserve f for Function of E,F;
reserve g for Function of F,G;
reserve a,b,c for Point of E;
reserve t for Real;

theorem Th24:
  f is isometric implies f is_continuous_on dom f
  proof
    assume
A1: f is isometric;
    set X = dom f;
    for s1 being sequence of E st rng s1 c= X & s1 is convergent &
    lim s1 in X holds f/*s1 is convergent & f/.(lim s1) = lim (f/*s1)
    proof
      let s1 be sequence of E;
      assume that
A2:   rng s1 c= X and
A3:   s1 is convergent and lim s1 in X;
      consider a such that
A4:   for r being Real st 0 < r
      ex m being Nat st for n being Nat st m <= n
      holds ||. s1.n - a .|| < r by A3;
A5:   a = lim s1 by A3,A4,NORMSP_1:def 7;
A6:   for r being Real st 0 < r ex m being Nat st
      for n being Nat st m <= n holds ||. (f/*s1).n - f.a .|| < r
      proof
        let r be Real;
        assume 0 < r;
        then consider m being Nat such that
A7:     for n being Nat st m <= n holds ||. s1.n - a .|| < r by A4;
        take m;
        let n be Nat;
A8:      n in NAT by ORDINAL1:def 12;
        assume m <= n;
        then
A9:     ||. s1.n - a .|| < r by A7;
A10:     ||. f.(s1.n) - f.a .|| = ||. s1.n - a .|| by A1;
        f/*s1 = f*s1 by A2,FUNCT_2:def 11;
        hence ||. (f/*s1).n - f.a .|| < r by A9,A10,FUNCT_2:15,A8;
      end;
      thus f/*s1 is convergent
      by A6;
      hence f/.(lim s1) = lim (f/*s1) by A5,A6,NORMSP_1:def 7;
    end;
    hence thesis by NFCONT_1:18;
  end;
