reserve x for Real,

  Z for open Subset of REAL;

theorem
  sin*cos is_differentiable_on Z & for x st x in Z holds((sin*cos)`|Z).x
  = -cos.(cos.x)*sin.x
proof
A1: for x st x in Z holds sin*cos is_differentiable_in x
  proof
    let x;
    assume x in Z;
A2: sin is_differentiable_in cos.x by SIN_COS:64;
    cos is_differentiable_in x by SIN_COS:63;
    hence thesis by A2,FDIFF_2:13;
  end;
  rng cos c= dom cos by SIN_COS:24;
  then
A3: dom (sin*cos) = REAL by RELAT_1:27,SIN_COS:24;
  then
A4: sin*cos is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds((sin*cos)`|Z).x = -cos.(cos.x)*sin.x
  proof
    let x;
    assume
A5: x in Z;
A6: sin is_differentiable_in cos.x by SIN_COS:64;
    cos is_differentiable_in x by SIN_COS:63;
    then diff(sin*cos,x) =diff(sin,cos.x)*diff(cos,x) by A6,FDIFF_2:13
      .=cos.(cos.x)*diff(cos,x) by SIN_COS:64
      .=cos.(cos.x)*(-sin.x) by SIN_COS:63;
    hence thesis by A4,A5,FDIFF_1:def 7;
  end;
  hence thesis by A3,A1,FDIFF_1:9;
end;
