reserve X for RealNormSpace;

theorem
  for X,Y be RealNormSpace, f be Function of TopSpaceNorm X,TopSpaceNorm
Y, ft be Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y st f=ft holds f
  is continuous iff ft is continuous
proof
  let X,Y be RealNormSpace, f be Function of TopSpaceNorm X,TopSpaceNorm Y, ft
  be Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y;
  assume
A1: f=ft;
  hereby
    assume
A2: f is continuous;
    now
      let xt be Point of LinearTopSpaceNorm X;
      reconsider x=xt as Point of TopSpaceNorm X by Def4;
      f is_continuous_at x by A2,TMAP_1:44;
      hence ft is_continuous_at xt by A1,Th35;
    end;
    hence ft is continuous by TMAP_1:44;
  end;
  assume
A3: ft is continuous;
  now
    let x be Point of TopSpaceNorm X;
    reconsider xt=x as Point of LinearTopSpaceNorm X by Def4;
    ft is_continuous_at xt by A3,TMAP_1:44;
    hence f is_continuous_at x by A1,Th35;
  end;
  hence thesis by TMAP_1:44;
end;
