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

theorem Th4:
  for T being non empty TopSpace holds T is compact iff for F
  being Subset-Family of T st F is centered & F is closed holds meet F <> {}
proof
  let T be non empty TopSpace;
  thus T is compact implies for F being Subset-Family of T st F is centered &
  F is closed holds meet F <> {}
  proof
    assume
A1: T is compact;
    let F be Subset-Family of T such that
A2: F is centered and
A3: F is closed;
    reconsider C = COMPLEMENT(F) as Subset-Family of T;
    assume
A4: meet F = {};
    F <> {} by A2,FINSET_1:def 3;
    then union COMPLEMENT(F) = (meet F)` by TOPS_2:7
      .= [#] T by A4;
    then
A5: COMPLEMENT(F) is Cover of T by SETFAM_1:def 11;
    COMPLEMENT(F) is open by A3,TOPS_2:9;
    then consider G9 being Subset-Family of T such that
A6: G9 c= C and
A7: G9 is Cover of T and
A8: G9 is finite by A1,A5;
    set F9= COMPLEMENT(G9);
A9: F9 is finite by A8,TOPS_2:8;
A10: F9 c= F
    proof
      let X be object;
      assume
A11:  X in F9;
      then reconsider X1=X as Subset of T;
      X1` in G9 by A11,SETFAM_1:def 7;
      then X1`` in F by A6,SETFAM_1:def 7;
      hence thesis;
    end;
    G9 <> {} by A7,TOPS_2:3;
    then
A12: F9 <> {} by TOPS_2:5;
    meet F9 = (union G9)` by A7,TOPS_2:3,6
      .= ([#] T) \ ([#] T) by A7,SETFAM_1:45
      .= {} by XBOOLE_1:37;
    hence contradiction by A2,A10,A9,A12,FINSET_1:def 3;
  end;
  assume
A13: for F being Subset-Family of T st F is centered & F is closed holds
  meet F <> {};
  thus T is compact
  proof
    let F be Subset-Family of T such that
A14: F is Cover of T and
A15: F is open;
    reconsider G=COMPLEMENT(F) as Subset-Family of T;
A16: G is closed by A15,TOPS_2:10;
    F <> {} by A14,TOPS_2:3;
    then
A17: G <> {} by TOPS_2:5;
    meet G = (union F)` by A14,TOPS_2:3,6
      .= ([#] T) \ ([#] T) by A14,SETFAM_1:45
      .= {} by XBOOLE_1:37;
    then not G is centered by A13,A16;
    then consider G9 being set such that
A18: G9 <> {} and
A19: G9 c= G and
A20: G9 is finite and
A21: meet G9 = {} by A17,FINSET_1:def 3;
    reconsider G9 as Subset-Family of T by A19,XBOOLE_1:1;
    take F9=COMPLEMENT(G9);
    thus F9 c= F
    proof
      let A be object;
      assume
A22:  A in F9;
      then reconsider A1=A as Subset of T;
      A1` in G9 by A22,SETFAM_1:def 7;
      then A1`` in F by A19,SETFAM_1:def 7;
      hence thesis;
    end;
    union F9 = (meet G9)` by A18,TOPS_2:7
      .= [#] T by A21;
    hence F9 is Cover of T by SETFAM_1:def 11;
    thus thesis by A20,TOPS_2:8;
  end;
end;
