reserve r for Real;
reserve a, b for Real;
reserve T for non empty TopSpace;
reserve A for non empty SubSpace of T;
reserve P,Q for Subset of T,
  p for Point of T;

theorem
  for X being TopStruct, Y being non empty TopStruct, K0 being Subset of
X, f being Function of X,Y, g being Function of X|K0,Y st f is continuous & g =
  f|K0 holds g is continuous
proof
  let X be TopStruct,Y be non empty TopStruct, K0 be Subset of X, f be
  Function of X,Y,g be Function of X|K0,Y;
  assume that
A1: f is continuous and
A2: g=f|K0;
A3: [#]Y <> {};
  for G being Subset of Y st G is open holds g"G is open
  proof
    let G be Subset of Y;
    [#](X|K0)=K0 by PRE_TOPC:def 5;
    then
A4: g"G= [#](X|K0) /\ f"G by A2,FUNCT_1:70;
    assume G is open;
    then f"G is open by A3,A1,TOPS_2:43;
    hence thesis by A4,TOPS_2:24;
  end;
  hence thesis by A3,TOPS_2:43;
end;
