
theorem Th13:
  for m being CR_Sequence, j being Nat, x be Integer st j in dom m
  holds (x mod Product(m)) mod m.j = x mod m.j
proof
  let m be CR_Sequence, j be Nat, x be Integer;
  assume
A1: j in dom m;
  consider z being Integer such that
A2: z * m.j = Product(m) by A1,INT_6:10;
  (x mod Product(m)) mod m.j = (x - (x div (z * m.j)) * (z * m.j)) mod m.j
  by A2,INT_1:def 10
    .= ((x mod m.j) - (0 + ((x div (z * m.j)) * z) * m.j mod m.j)) mod m.j
      by INT_6:7
    .= ((x mod m.j) - (0 mod m.j)) mod m.j by NAT_D:61
    .= (x - 0) mod m.j by INT_6:7
    .= x mod m.j;
  hence thesis;
end;
