
theorem Th41:
for P be set, F be FinSequence st
 P is cup-closed & {} in P & (for n be Nat st n in dom F holds F.n in P)
holds Union F in P
proof
   let P be set, F be FinSequence;
   assume that
A0: P is cup-closed and
A1: {} in P and
A2: for n be Nat st n in dom F holds F.n in P;
   defpred P[Nat] means union rng (F|$1) in P;
A3:P[0] by A1,ZFMISC_1:2;
A4:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A5: P[k];
A6: k <= k+1 by NAT_1:13;
    per cases;
    suppose A7: len F >= k+1; then
     len (F|(k+1)) = k+1 by FINSEQ_1:59; then
     F|(k+1) = ((F|(k+1))|k) ^ <* (F|(k+1)).(k+1) *> by FINSEQ_3:55
      .= F|k ^ <* (F|(k+1)).(k+1) *> by A6,FINSEQ_1:82
      .= F|k ^ <* F.(k+1) *> by FINSEQ_3:112; then
     rng(F|(k+1)) = rng(F|k) \/ rng <* F.(k+1) *> by FINSEQ_1:31
      .= rng(F|k) \/ {F.(k+1)} by FINSEQ_1:38; then
A8:  union rng(F|(k+1)) = union rng(F|k) \/ union {F.(k+1)} by ZFMISC_1:78
      .= union rng(F|k) \/ F.(k+1) by ZFMISC_1:25;
     1 <= k+1 by NAT_1:11; then
     F.(k+1) in P by A2,A7,FINSEQ_3:25;
     hence P[k+1] by A0,A5,A8,FINSUB_1:def 1;
    end;
    suppose len F < k+1; then
     (F|(k+1)) = F & F|k = F by FINSEQ_3:49,NAT_1:13;
     hence P[k+1] by A5;
    end;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A3,A4); then
   union rng (F| len F) in P; then
   union rng F in P by FINSEQ_3:49;
   hence thesis by CARD_3:def 4;
end;
