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

theorem
  for F being integer-valued Function holds
  (a(#)(F mod m)) mod m = (a(#)F) mod m
proof
  let F be integer-valued Function;
A3: dom (a(#)F) = dom F by VALUED_1:def 5;
    dom (a(#)(F mod m)) = dom(F mod m) by VALUED_1:def 5;
    then
A4: dom (a(#)(F mod m)) = dom F by Def1;
A5: for x being object st x in dom F
   holds ((a(#)(F mod m)) mod m).x = ((a(#)F) mod m).x
  proof
    let x be object;
    assume
A6: x in dom F;
A1: (a(#)(F mod m)).x = a*(F mod m).x by VALUED_1:6;
    ((a(#)(F mod m)) mod m).x = (a(#)(F mod m)).x mod m by A4,A6,Def1
      .= (a*(F.x mod m)) mod m by A6,A1,Def1
      .= (a*F.x) mod m by Th7
      .= (a(#)F).x mod m by VALUED_1:6
      .= ((a(#)F) mod m).x by A3,A6,Def1;
    hence thesis;
  end;
A7: dom ((a(#)F) mod m) = dom F by A3,Def1;
  dom ((a(#)(F mod m)) mod m) = dom F by A4,Def1;
  hence thesis by A7,A5;
end;
