
theorem
  for m being CR_Sequence, X be Group-Sequence st len m = len X &
  (for i be Element of NAT st i in dom X holds ex mi be non zero Nat
  st mi = m.i & X.i = Z/Z (mi)) holds
  ex I being Homomorphism of Z/Z (Product(m)),product X st I is bijective &
  (for x be Integer st x in the carrier of Z/Z (Product(m)) holds
  I.x = mod(x,m))
proof
  let m be CR_Sequence, X be Group-Sequence;
  assume
A1: len m = len X & (for i be Element of NAT st i in dom X holds ex mi be
  non zero Nat st mi = m.i & X.i = Z/Z (mi));
  then consider I be Homomorphism of Z/Z (Product(m)),product X such that
A2: (for x be Integer st x in the carrier of Z/Z (Product(m)) holds I.x =
  mod(x,m)) by Th14;
A3: I is one-to-one by A1,Th20,A2;
  Product(m) is Nat by TARSKI:1; then
A4: card(Segm Product(m)) = Product(m);
A5: card(the carrier of product X) = Product(m) by A1,Th19;
  then the carrier of product X is finite;
  then I is onto by A3,A4,A5,FINSEQ_4:63;
  hence thesis by A2,A3;
end;
