
theorem
  for m being CR_Sequence, c1,c2 being CR_coefficients of m, i being
  Nat st i in dom c1 holds c1.i,c2.i are_congruent_mod m.i
proof
  let m be CR_Sequence, c1,c2 be CR_coefficients of m, i be Nat;
  assume
A1: i in dom c1;
  then
A2: ex s1 being Integer, mm1 being Integer st mm1 = Product(m ) / m.i & s1
  * mm1, 1 are_congruent_mod m.i & c1.i = s1 * (Product(m) / m.i) by Def4;
A3: len c1 = len m by Def4
    .= len c2 by Def4;
  dom c1 = Seg(len c1) by FINSEQ_1:def 3
    .= dom c2 by A3,FINSEQ_1:def 3;
  then consider s2 being Integer, mm2 being Integer such that
A4: mm2 = Product(m) / m.i and
A5: s2 * mm2, 1 are_congruent_mod m.i and
A6: c2.i = s2 * (Product(m) / m.i) by A1,Def4;
  1, s2 * mm2 are_congruent_mod m.i by A5,INT_1:14;
  hence thesis by A2,A4,A6,INT_1:15;
end;
