 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th3:
for S be non empty TopSpace,T be NormedLinearTopSpace,
    f be Function of S,T,
    x be Point of S holds
(  f is_continuous_at x
iff
 for e be Real
     st 0 < e
 holds ex H being Subset of S st
( H is open & x in H
&
for y be Point of S
   st y in H holds ||.f.x-f.y.|| < e ) )
proof
let S be non empty TopSpace,T be NormedLinearTopSpace,
    f be Function of S,T,
    x be Point of S;
hereby assume
A1:f is_continuous_at x;
  let e be Real;
  assume A2: 0 < e;
  set V = { y where y is Point of T : ||.f.x - y.|| < e };
  now let z be object;
    assume z in V; then
    ex y be Point of T st z=y & ||.f.x - y.|| < e;
    hence z in the carrier of T;
 end; then
 reconsider V as Subset of T by TARSKI:def 3;
 ||.f.x - f.x.|| < e by NORMSP_1:6,A2; then
f.x in V; then
consider H being Subset of S such that
A3:  H is open & x in H & f .: H c= V by A1,TMAP_1:43,C0SP3:24;
take H;
  thus H is open by A3;
  thus x in H by A3;
  let y be Point of S;
  assume y in H; then
  f.y in f.:H by FUNCT_2:35; then
  f.y in V by A3; then
  ex z be Point of T st z=f.y & ||.f.x - z.|| < e;
  hence ||.f.x - f.y.|| < e;
end;
assume
A4:  for e be Real st 0 < e
 holds ex H being Subset of S st
( H is open & x in H & for y be Point of S
   st y in H holds ||.f.x-f.y.|| < e );
for G being Subset of T st G is open & f . x in G holds
ex H being Subset of S st
  H is open & x in H & f .: H c= G
proof
   let G be Subset of T;
   assume G is open & f . x in G; then
   consider r being Real such that
   A5:  r > 0
   & { y where y is Point of T : ||. f.x - y .|| < r } c= G by C0SP3:23;
  consider H being Subset of S such that
   A6: H is open & x in H &
      for y be Point of S
      st y in H holds ||.f.x-f.y.|| < r by A4,A5;
  take H;
  thus H is open by A6;
  thus x in H by A6;
  now let z be object;
    assume z in f .: H; then
    consider t being object such that
    A7: t in dom f & t in H & z = f . t by FUNCT_1:def 6;
    reconsider t as Point of S by A7;
    ||.f.x-f.t.|| < r by A7,A6; then
    z in { y where y is Point of T : ||. f.x - y .|| < r } by A7;
    hence z in G by A5;
 end;
 hence f .: H c= G;
end;
hence f is_continuous_at x by TMAP_1:43;
end;
