reserve X for RealNormSpace;

theorem Th18:
  for X,Y be RealNormSpace, f be PartFunc of X,Y, ft be Function
of TopSpaceNorm X,TopSpaceNorm Y, x be Point of X, xt be Point of TopSpaceNorm
  X st f = ft & x = xt holds f is_continuous_in x iff ft is_continuous_at xt
proof
  let X,Y be RealNormSpace, f be PartFunc of X,Y, ft be Function of
TopSpaceNorm X,TopSpaceNorm Y, x be Point of X, xt be Point of TopSpaceNorm X;
  assume that
A1: f = ft and
A2: x = xt;
A3: dom f = the carrier of X by A1,FUNCT_2:def 1;
  then
A4: ft.xt = f/.x by A1,A2,PARTFUN1:def 6;
  hereby
    assume
A5: f is_continuous_in x;
    now
      let G be a_neighborhood of ft.xt;
      reconsider N1 = G as Subset of Y;
      N1 is Neighbourhood of f/.x by A4,Th17;
      then consider N being Neighbourhood of x such that
A6:   f.:N c= N1 by A5,NFCONT_1:10;
      reconsider H=N as a_neighborhood of xt by A2,Th17;
      take H;
      thus ft.:H c= G by A1,A6;
    end;
    hence ft is_continuous_at xt by TMAP_1:def 2;
  end;
  assume
A7: ft is_continuous_at xt;
  now
    let N1 be Neighbourhood of f/.x;
    reconsider G=N1 as Subset of Y;
    G is a_neighborhood of ft.xt by A4,Th17;
    then consider H being a_neighborhood of xt such that
A8: ft.:H c= G by A7,TMAP_1:def 2;
    reconsider N=H as Subset of X;
    reconsider N as Neighbourhood of x by A2,Th17;
    take N;
    thus f.:N c= N1 by A1,A8;
  end;
  hence thesis by A3,NFCONT_1:10;
end;
