
theorem Th27:
  for m being CR_Sequence, c being CR_coefficients of m, i,j being
  Nat st i in dom c & j in dom c & i <> j holds c.i,0
  are_congruent_mod m.j
proof
  let m be CR_Sequence, c be CR_coefficients of m, i,j be Nat;
  assume that
A1: i in dom c and
A2: j in dom c and
A3: i <> j;
  consider s being Integer, mm being Integer such that
A4: mm = Product(m) / m.i and
  s * mm, 1 are_congruent_mod m.i and
A5: c.i = s * (Product(m) / m.i) by A1,Def4;
  len m = len c by Def4;
  then dom m = Seg(len c) by FINSEQ_1:def 3
    .= dom c by FINSEQ_1:def 3;
  then consider z being Integer such that
A6: z * m.j = mm by A2,A3,A4,Lm6;
A7: m.j, 0 are_congruent_mod m.j by INT_1:12;
A8: s,s are_congruent_mod m.j by INT_1:11;
  z,z are_congruent_mod m.j by INT_1:11;
  then z * m.j, z*0 are_congruent_mod m.j by A7,INT_1:18;
  then s * mm, s * 0 are_congruent_mod m.j by A6,A8,INT_1:18;
  hence thesis by A4,A5;
end;
