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

theorem Th27:
  for a1,a2 being Integer
  for n1,n2 being Nat st n1 > 0 & n2 > 0
  for x being Integer st x solves_CRT a1,n1,a2,n2 holds
  x mod (n1*n2) solves_CRT a1,n1,a2,n2
  proof
    let a1,a2 be Integer;
    let n1,n2 be Nat such that
A1: n1 > 0 and
A2: n2 > 0;
    let x be Integer such that
A3: x solves_CRT a1,n1,a2,n2;
A4: x mod n1 = a1 mod n1 by A3,NAT_D:64;
    (x mod (n1*n2)) mod n1 = x mod n1 by A2,RADIX_1:7;
    hence x mod (n1*n2),a1 are_congruent_mod n1 by A1,A4,NAT_D:64;
A5: x mod n2 = a2 mod n2 by A3,NAT_D:64;
    (x mod (n1*n2)) mod n2 = x mod n2 by A1,RADIX_1:7;
    hence x mod (n1*n2),a2 are_congruent_mod n2 by A2,A5,NAT_D:64;
  end;
