
theorem
  for f,g be Function of NAT,REAL
  holds f negligibleEQ g iff f-g is negligible
  proof
    let f,g be Function of NAT,REAL;
    hereby assume
      A1: f negligibleEQ g;
      consider v be Function of NAT,REAL such that
      B1: v is negligible &
      for x be Nat
      holds |. f.x - g.x .| <= |.v.x.| by A1;
      for x be Nat holds |. (f-g).x .| <= |.v.x.|
      proof
        let x be Nat;
        x in NAT by ORDINAL1:def 12;then
        consider y be Element of NAT such that LXY: x=y;
        |. (f-g).y .| = |. f.y-g.y .| by VALUED_1:15;
        hence thesis by B1,LXY;
      end;
      hence (f-g) is negligible by B1,TH4;
    end;
    set v = f-g;
    for x be Nat holds |. f.x-g.x .| <= |.v.x.|
    proof
      let x be Nat;
      x in NAT by ORDINAL1:def 12;
      hence thesis by VALUED_1:15;
    end;
    hence thesis;
  end;
