reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;
reserve p,q for natural Number;
reserve i0,i,i1,i2,i4 for Integer;

theorem
  0 < m implies n mod m = n - m * (n div m)
proof
  reconsider z1=m * (n div m),z2=(n mod m) as Integer;
  assume m > 0;
  then n - z1 = z1 + z2 -z1 by NAT_D:2;
  hence thesis;
end;
