reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem Th10:
  (id Z)(#)cos is_differentiable_on Z & for x st x in Z holds (((
  id Z)(#)cos)`|Z).x =cos.x-x*sin.x
proof
A1: cos is_differentiable_on Z by FDIFF_1:26,SIN_COS:67;
A2: dom ((id Z)(#)cos) = dom (id Z) /\ REAL by SIN_COS:24,VALUED_1:def 4
    .= dom (id Z) by XBOOLE_1:28
    .= Z by RELAT_1:45;
  then Z c= dom (id Z) /\ dom cos by VALUED_1:def 4;
  then
A3: Z c= dom (id Z) by XBOOLE_1:18;
  for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  then
A4: id Z is_differentiable_on Z by A3,FDIFF_1:23;
A5: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  now
    let x;
    assume
A6: x in Z;
    hence
    (((id Z)(#)cos)`|Z).x = (cos.x)*diff((id Z),x) + ((id Z).x)*diff(cos,
    x) by A2,A4,A1,FDIFF_1:21
      .=(cos.x)*((id Z)`|Z).x + ((id Z).x)*diff(cos,x) by A4,A6,FDIFF_1:def 7
      .=(cos.x)*1+ ((id Z).x)*diff(cos,x) by A3,A5,A6,FDIFF_1:23
      .=(cos.x)*1+((id Z).x)*(-sin.x) by SIN_COS:63
      .=cos.x+x*(-sin.x) by A6,FUNCT_1:18
      .=cos.x-x*sin.x;
  end;
  hence thesis by A2,A4,A1,FDIFF_1:21;
end;
