reserve X for RealNormSpace;

theorem Th35:
  for X,Y be RealNormSpace, f be Function of TopSpaceNorm X,
TopSpaceNorm Y, ft be Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y, x
be Point of TopSpaceNorm X, xt be Point of LinearTopSpaceNorm X st f=ft & x=xt
  holds f is_continuous_at x iff ft is_continuous_at xt
proof
  let X,Y be RealNormSpace, f be Function of TopSpaceNorm X,TopSpaceNorm Y, ft
  be Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y, x be Point of
  TopSpaceNorm X, xt be Point of LinearTopSpaceNorm X;
  assume that
A1: f=ft and
A2: x=xt;
  hereby
    assume
A3: f is_continuous_at x;
    now
      let Gt be a_neighborhood of ft.xt;
      Gt is Subset of TopSpaceNorm Y by Def4;
      then reconsider G=Gt as a_neighborhood of f.x by A1,A2,Th34;
      consider H being a_neighborhood of x such that
A4:   f.:H c= G by A3,TMAP_1:def 2;
      H is Subset of LinearTopSpaceNorm X by Def4;
      then reconsider Ht=H as a_neighborhood of xt by A2,Th34;
      take Ht;
      thus ft.:Ht c= Gt by A1,A4;
    end;
    hence ft is_continuous_at xt by TMAP_1:def 2;
  end;
  assume
A5: ft is_continuous_at xt;
  now
    let G be a_neighborhood of f.x;
    G is Subset of LinearTopSpaceNorm Y by Def4;
    then reconsider Gt=G as a_neighborhood of ft.xt by A1,A2,Th34;
    consider Ht being a_neighborhood of xt such that
A6: ft.:Ht c= Gt by A5,TMAP_1:def 2;
    Ht is Subset of TopSpaceNorm X by Def4;
    then reconsider H=Ht as a_neighborhood of x by A2,Th34;
    take H;
    thus f.:H c= G by A1,A6;
  end;
  hence thesis by TMAP_1:def 2;
end;
