
theorem Th5:
  for X being TopStruct holds X is SubSpace of One-Point_Compactification(X)
proof
  let X be TopStruct;
  set D = {U \/ {[#]X} where U is Subset of X: U is open & U` is compact};
  thus
A1: [#]X c= [#]One-Point_Compactification(X) by Th4;
  let P being Subset of X;
A2: the topology of One-Point_Compactification(X) = (the topology of X) \/ D
  by Def9;
  hereby
    reconsider Q = P as Subset of One-Point_Compactification(X) by A1,
XBOOLE_1:1;
    assume
A3: P in the topology of X;
    take Q;
    thus Q in the topology of One-Point_Compactification(X) by A2,A3,
XBOOLE_0:def 3;
    thus P = Q /\ [#]X by XBOOLE_1:28;
  end;
  given Q being Subset of One-Point_Compactification(X) such that
A4: Q in the topology of One-Point_Compactification(X) and
A5: P = Q /\ [#]X;
  per cases by A2,A4,XBOOLE_0:def 3;
  suppose
    Q in the topology of X;
    hence thesis by A5,XBOOLE_1:28;
  end;
  suppose
    Q in D;
    then consider U being Subset of X such that
A6: Q = U \/ {[#]X} and
A7: U is open and
    U` is compact;
    not [#]X in [#]X;
    then {[#]X} misses [#]X by ZFMISC_1:50;
    then {[#]X} /\ [#]X = {};
    then P = (U /\ [#]X) \/ {} by A5,A6,XBOOLE_1:23
      .= U by XBOOLE_1:28;
    hence thesis by A7;
  end;
end;
