reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th30:
  for u being integer-valued FinSequence
  for m being CR_Sequence st dom u = dom m holds
  for z being Integer st z solves_CRT u,m
  for k being Integer holds z + k*Product(m) solves_CRT u,m
  proof
    let u be integer-valued FinSequence;
    let m be CR_Sequence such that
A1: dom u = dom m;
    let z be Integer such that
A2: z solves_CRT u,m;
    let k be Integer;
    let i be Nat such that
A3: i in dom u;
A4: z,u.i are_congruent_mod m.i by A2,A3;
A5: m.i in rng m by A1,A3,FUNCT_1:def 3;
    then m.i divides Product(m) by LIOUVIL2:22;
    then Product(m) mod m.i = 0 by A5,INT_1:62;
    then (k*Product(m)) mod m.i = ((k mod m.i) * 0) mod m.i by NAT_D:67
    .= 0 mod m.i;
    then k*Product(m),0 are_congruent_mod m.i by A1,A3,NAT_D:64;
    then z+k*Product(m),u.i+0 are_congruent_mod m.i by A4,INT_1:16;
    hence z+k*Product(m),u.i are_congruent_mod m.i;
  end;
