reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem
  for T being TopSpace, F being finite Subset-Family of T st for X being
  Subset of T st X in F holds X is compact holds union F is compact
proof
  let T be TopSpace, F be finite Subset-Family of T such that
A1: for X being Subset of T st X in F holds X is compact;
  defpred P[set] means ex A being Subset of T st A = union $1 & A is compact;
A2: for x, B being set st x in F & B c= F & P[B] holds P[B \/ {x}]
  proof
    let x, B be set such that
A3: x in F and
A4: B c= F;
    B c= bool the carrier of T
    proof
      let b be object;
      assume b in B;
      then b in F by A4;
      hence thesis;
    end;
    then reconsider C = B as Subset-Family of T;
    reconsider X = x as Subset of T by A3;
    given A being Subset of T such that
A5: A = union B and
A6: A is compact;
    set K = union C \/ X;
    take K;
    union {x} = x by ZFMISC_1:25;
    hence K = union (B \/ {x}) by ZFMISC_1:78;
    X is compact by A1,A3;
    hence thesis by A5,A6,COMPTS_1:10;
  end;
A7: P[{}]
  proof
    take {}T;
    thus thesis by ZFMISC_1:2;
  end;
A8: F is finite;
  P[F] from FINSET_1:sch 2(A8,A7,A2);
  hence thesis;
end;
