
theorem Th32:
  for u being Integer, m being CR_Sequence, i being Nat
  st i in dom m holds u, mod(u,m).i are_congruent_mod m.i
proof
  let u be Integer, m be CR_Sequence;
  let i be Nat;
A1: len mod(u,m) = len m by Def3;
  assume
A2: i in dom m;
  then m.i in rng m by FUNCT_1:3;
  then m.i > 0 by PARTFUN3:def 1;
  then u mod m.i = u - (u div m.i) * m.i by INT_1:def 10;
  then
A3: u - (u mod m.i) = (u div m.i) * m.i;
  dom mod(u,m) = Seg(len mod(u,m)) by FINSEQ_1:def 3
    .= dom m by A1,FINSEQ_1:def 3;
  then mod(u,m).i = u mod m.i by A2,Def3;
  hence thesis by A3,INT_1:def 5;
end;
