
theorem
  for Y being non empty TopSpace, W being non empty SubSpace of Y holds
  corestr incl W is being_homeomorphism
proof
  let Y be non empty TopSpace, W be non empty SubSpace of Y;
  set ci = corestr incl W;
  thus dom ci = [#]W by FUNCT_2:def 1;
  thus
A1: rng ci = [#]Image incl W by FUNCT_2:def 3;
A2: ci = incl W by WAYBEL18:def 7
    .= (id Y)|W by TMAP_1:def 6
    .= (id the carrier of Y)|the carrier of W by TMAP_1:def 4;
  hence
A3: ci is one-to-one by FUNCT_1:52;
A4: for P being Subset of W st P is open holds ci/""P is open
  proof
    let P be Subset of W;
    assume P in the topology of W;
    then
A5: ex Q being Subset of Y st Q in the topology of Y & P = Q /\ [#]W by
PRE_TOPC:def 4;
A6: the carrier of W is non empty Subset of Y by BORSUK_1:1;
    then
A7: P is Subset of Y by XBOOLE_1:1;
A8: [#]W = rng ((id the carrier of Y)|the carrier of W) by A6,Th2
      .= rng ((id Y)|W) by TMAP_1:def 4
      .= rng incl W by TMAP_1:def 6
      .= [#]Image incl W by WAYBEL18:9;
    ci/""P = ((id the carrier of Y)|the carrier of W).:P by A1,A2,A3,TOPS_2:54
      .= (id the carrier of Y).:P by FUNCT_2:97
      .= P by A7,FUNCT_1:92;
    hence ci/""P in the topology of Image incl W by A5,A8,PRE_TOPC:def 4;
  end;
  thus ci is continuous by WAYBEL18:10;
  [#]W <> {};
  hence thesis by A4,TOPS_2:43;
end;
