reserve a,b,m,x,n,l,xi,xj for Nat,
  t,z for Integer;

theorem
  for F being integer-valued Function holds (F mod m) mod m = F mod m
proof
  let F be integer-valued Function;
A1: dom (F mod m) = dom F by Def1;
A2: for x being object st x in dom F holds ((F mod m) mod m).x = (F mod m).x
  proof
    let x be object;
    assume
A3: x in dom F;
    ((F mod m) mod m).x = ((F mod m).x) mod m by A1,A3,Def1
      .= (F.x mod m) mod m by A3,Def1
      .= F.x mod m by NAT_D:73
      .= (F mod m).x by A3,Def1;
    hence thesis;
  end;
  dom ((F mod m) mod m) = dom (F mod m) by Def1;
  hence thesis by A1,A2;
end;
