reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;

theorem Th8:
  T is compact & P is closed implies P is compact
proof
  assume that
A1: T is compact and
A2: P is closed;
  reconsider pp = {P`} as Subset-Family of T;
  let F be Subset-Family of T such that
A3: F is Cover of P and
A4: F is open;
  set FP = F \/ pp;
A5: P` is open by A2;
A6: FP is open
  proof
    let H be Subset of T;
    assume H in FP;
    then H in F or H in {P`} by XBOOLE_0:def 3;
    hence thesis by A4,A5,TARSKI:def 1;
  end;
  reconsider M = {P`} as Subset-Family of T;
  (F \/ {P`}) \ {P`} = F \ {P`} by XBOOLE_1:40;
  then
A7: (F \/ {P`}) \ {P`} c= F by XBOOLE_1:36;
  P c= union F by A3,SETFAM_1:def 11;
  then P \/ (P`) c= (union F) \/ (P`) by XBOOLE_1:9;
  then [#] T c= (union F) \/ (P`) by PRE_TOPC:2;
  then [#] T = (union F) \/ (P`)
    .= (union F) \/ (union {P`}) by ZFMISC_1:25
    .= union (F \/ {P`}) by ZFMISC_1:78;
  then FP is Cover of T by SETFAM_1:def 11;
  then consider G being Subset-Family of T such that
A8: G c= FP and
A9: G is Cover of T and
A10: G is finite by A1,A6;
  reconsider G9 = G \ pp as Subset-Family of T;
  take G9;
  G9 c= (F \/ {P`}) \ {P`} by A8,XBOOLE_1:33;
  hence G9 c= F by A7;
  (union G) \ union M = [#] T \ (union {P`}) by A9,SETFAM_1:45
    .= P`` by ZFMISC_1:25
    .= P;
  then P c= union G9 by TOPS_2:4;
  hence G9 is Cover of P by SETFAM_1:def 11;
  thus thesis by A10;
end;
