 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th2:
for S be TopSpace,
    F be FinSequence of bool the carrier of S
    st for i be Nat st i in Seg len F
        holds F/.i is compact
holds
  union rng F is compact
proof
  let S be TopSpace;
  let F be FinSequence of bool the carrier of S;
   assume A1: for i be Nat st i in Seg len F holds F/.i is compact;
  defpred P[Nat] means
  for F be FinSequence of bool the carrier of S
    st len F = $1 &
       for i be Nat st i in Seg len F
        holds F/.i is compact holds union rng F is compact;
A2:P[0]
proof
let F be FinSequence of bool the carrier of S;
  assume A3:len F = 0 &
       for i be Nat st i in Seg len F holds F/.i is compact;
  dom F = {} by A3,FINSEQ_1:def 3; then
union rng F = {} by ZFMISC_1:2,RELAT_1:42;
hence union rng F is compact;
end;
A4:for i be Nat st P[i] holds P[i+1]
proof
  let i be Nat;
  assume A5: P[i];
  let F be FinSequence of bool the carrier of S;
  assume A6:len F = i+1 &
       for i be Nat st i in Seg len F holds F/.i is compact;
  i+1 in Seg (i+1) by FINSEQ_1:4; then
A7: i+1 in dom F by A6,FINSEQ_1:def 3;
A8: F = (F|i) ^ <*F.(i+1)*> by A6,FINSEQ_3:55;
A9:i <= i + 1 by NAT_1:11;
A10: len (F|i) = i by A6,FINSEQ_1:59,NAT_1:11; then
A11:dom (F|i) = Seg i by FINSEQ_1:def 3;
  for k be Nat st k in Seg len (F|i) holds (F|i)/.k is compact
proof
  let k be Nat;
   assume
   A12: k in Seg len (F|i); then
   A13: k in Seg (i+1) by A10,A9,FINSEQ_1:5,TARSKI:def 3; then
A14: k in dom F by A6,FINSEQ_1:def 3;
  (F|i)/.k = (F|i).k by A11,A12,A10,PARTFUN1:def 6
               .=F.k by A12,A10,FUNCT_1:49
               .=F/.k by PARTFUN1:def 6,A14;
   hence (F|i)/.k is compact by A6,A13;
end; then
A15: union rng (F|i) is compact by A5,A10;
rng F = rng (F|i) \/ rng <*F.(i+1)*> by FINSEQ_1:31,A8
     .= rng (F|i) \/ {F.(i+1) } by FINSEQ_1:39
     .= rng (F|i) \/ {F/.(i+1) } by PARTFUN1:def 6,A7; then
A16:union rng F = (union rng (F|i)) \/ union {F/.(i+1) } by ZFMISC_1:78
               .= (union rng ( F|i )) \/ (F/.(i+1)) by ZFMISC_1:25;
F/.(i+1) is compact by A6,FINSEQ_1:4;
hence thesis by A16,A15,COMPTS_1:10;
end;
for n be Nat holds P[n] from NAT_1:sch 2(A2,A4);
hence thesis by A1;
end;
