
theorem Th13:
  for n being non zero Nat, x being Integer
  holds x,0 are_congruent_mod n or ... or x,(n-1) are_congruent_mod n
proof
  let n be non zero Nat, x be Integer;
  x mod n in NAT by INT_1:3,INT_1:57;
  then reconsider j = x mod n as Nat;
  (x mod n) + 1 <= n by INT_1:7,INT_1:58;
  then A1: (x mod n) + 1 - 1 <= n - 1 by XREAL_1:9;
  take j; thus thesis by Th10,A1;
end;
