
theorem Th34:
  for u,v being Integer, m being CR_Sequence, i being Nat
  st i in dom m holds (mod(u,m) (#) mod(v,m)).i, u * v are_congruent_mod m.i
proof
  let u,v be Integer, m be CR_Sequence, i be Nat;
  assume
A1: i in dom m;
A2: len mod(v,m) = len m by Def3;
  then dom mod(v,m) = Seg(len m) by FINSEQ_1:def 3
    .= dom m by FINSEQ_1:def 3;
  then
A3: mod(v,m).i = v mod m.i by A1,Def3;
A4: len mod(u,m) = len m by Def3;
  then len(mod(u,m)(#)mod(v,m)) = len m by A2,Lm4;
  then dom(mod(u,m)(#)mod(v,m)) = Seg(len m) by FINSEQ_1:def 3
    .= dom m by FINSEQ_1:def 3;
  then
A5: (mod(u,m)(#)mod(v,m)).i = ((mod(u,m).i) * (mod(v,m).i)) by A1,
VALUED_1:def 4;
  dom mod(u,m) = Seg(len m) by A4,FINSEQ_1:def 3
    .= dom m by FINSEQ_1:def 3;
  then mod(u,m).i = u mod m.i by A1,Def3;
  then
A6: (mod(u,m).i * mod(v,m).i) mod m.i = (u * v) mod m.i by A3,NAT_D:67;
  m.i in rng m by A1,FUNCT_1:3;
  then m.i > 0 by PARTFUN3:def 1;
  hence thesis by A5,A6,NAT_D:64;
end;
