reserve y for set;
reserve x,a,b,c for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th44:
  Z c= dom ((-id Z)(#)cos) implies (-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: for x st x in Z holds (-id Z).x =(-1)*x +0
  proof
    let x;
    assume
A2: x in Z;
    (-id Z).x = -(id Z.x) by VALUED_1:8
      .=-x by A2,FUNCT_1:18
      .=(-1)*x +0;
    hence thesis;
  end;
  assume
A3: Z c= dom ((-id Z)(#)cos);
  then Z c= dom (-id Z) /\ dom cos by VALUED_1:def 4;
  then
A4: Z c= dom (-id Z) by XBOOLE_1:18;
  then
A5: -id Z is_differentiable_on Z by A1,FDIFF_1:23;
A6: cos is_differentiable_on Z by FDIFF_1:26,SIN_COS:67;
  now
    let x;
    assume
A7: x in Z;
    hence (((-id Z)(#)cos)`|Z).x = (cos.x)*diff((-id Z),x) + ((-id Z).x)*diff(
    cos,x) by A3,A5,A6,FDIFF_1:21
      .=(cos.x)*((-id Z)`|Z).x+ ((-id Z).x)*diff(cos,x) by A5,A7,FDIFF_1:def 7
      .=(cos.x)*(-1)+ ((-id Z).x)*diff(cos,x) by A4,A1,A7,FDIFF_1:23
      .=(cos.x)*(-1)+((-id Z).x)*(-sin.x) by SIN_COS:63
      .=-cos.x+((-1)*x +0)*(-sin.x) by A1,A7
      .=-cos.x+x*sin.x;
  end;
  hence thesis by A3,A5,A6,FDIFF_1:21;
end;
