reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

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;
