
theorem Th10:
  for X, Y being non empty TopSpace, f being Function of X,Y st f
  is continuous holds corestr f is continuous
proof
  let X, Y be non empty TopSpace;
  let f be Function of X,Y;
A1: f is Function of dom f,rng f by FUNCT_2:1;
A2: [#]Y <> {};
  assume
A3: f is continuous;
A4: for R being Subset of Image f st R is open holds (corestr f)"R is open
  proof
    [#](Image f) = rng f by Th9;
    then
A5: f"([#](Image f)) = dom f by A1,FUNCT_2:40
      .= the carrier of X by FUNCT_2:def 1;
    the carrier of X in the topology of X by PRE_TOPC:def 1;
    then
A6: f"([#](Image f)) is open by A5;
    let R be Subset of Image f;
    assume R is open;
    then R in the topology of Image f;
    then consider Q being Subset of Y such that
A7: Q in the topology of Y and
A8: R = Q /\ [#](Image f) by PRE_TOPC:def 4;
    reconsider Q as Subset of Y;
    Q is open by A7;
    then
A9: f"Q is open by A3,A2,TOPS_2:43;
    f"Q /\ f"([#](Image f)) = f"(Q /\ [#](Image f)) by FUNCT_1:68;
    hence thesis by A8,A9,A6,TOPS_1:11;
  end;
  [#]Image f <> {};
  hence thesis by A4,TOPS_2:43;
end;
