reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem
  for n being natural Number, a being Integer holds (a mod n) mod n = a mod n
proof
  let n be natural Number;
  let a be Integer;
  per cases;
  suppose
A1: n = 0;
    hence (a mod n) mod n = 0 by INT_1:def 10
      .= a mod n by A1,INT_1:def 10;
  end;
  suppose
    n <> 0;
    then a mod n >= 0 & a mod n < n by Th62;
    hence thesis by Th63;
  end;
end;
