theorem
  for m,n,k being Nat holds 0 < n & m mod n = k implies n divides (m - k)
proof
  let m,n,k be Nat;
  assume
A1: 0 < n & m mod n = k;
  take m div n;
  m = n*(m div n) + k by A1,NAT_D:2;
  hence thesis;
end;
