
theorem Th9:
  for m being integer-valued FinSequence, i being Nat holds
  Product(m) / m.i is Integer
proof
  let m be integer-valued FinSequence, i be Nat;
  reconsider m2 = m/^i as FinSequence of INT;
  reconsider m1 = m|i as FinSequence of INT by FINSEQ_1:102;
  reconsider m9 = m as FinSequence of INT by FINSEQ_1:102;
  per cases;
  suppose not i in dom m or m.i = 0;
    then m.i = 0 by FUNCT_1:def 2;
    hence thesis;
  end;
  suppose that
A1: i in dom m and
A2: m.i <> 0;
A3: dom m = Seg(len m) by FINSEQ_1:def 3;
  then 1 <= i by A1,FINSEQ_1:1;
  then reconsider j = i - 1 as Element of NAT by INT_1:3;
  set f = (m|j) ^ (m/^i);
A4: m9 = m1 ^ m2 by RFINSEQ:8;
  j + 1 <= len m by A1,A3,FINSEQ_1:1;
  then (m|j) ^ <*m.i*> = (m|i) by Th5;
  then Product(m) = Product(m|j ^ <*m.i*>) * Product(m/^i) by A4,RVSUM_1:97
    .= Product(m|j) * Product(<*m.i*>) * Product(m/^i) by RVSUM_1:97
    .= (Product(m|j) * Product(m/^i)) * Product(<*m.i*>)
    .= Product(m|j) * Product(m/^i) * m.i
    .= Product(f) * m.i by RVSUM_1:97;
  then m.i divides Product(m) by INT_1:def 3;
  hence thesis by A2,WSIERP_1:17;
end;
end;
