reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem
  x in Z implies (diff(sin,Z).2).x = -sin.x
proof
  assume
A1: x in Z;
A2: cos is_differentiable_on Z by FDIFF_1:26,SIN_COS:67;
A3: sin is_differentiable_on Z by FDIFF_1:26,SIN_COS:68;
  (diff(sin,Z).(2*1)).x = (diff(sin,Z).(1 +1) ).x
    .=(diff(sin,Z).(1+0)`|Z).x by TAYLOR_1:def 5
    .=((diff(sin,Z).0 `|Z)`|Z).x by TAYLOR_1:def 5
    .=((sin | Z`|Z)`|Z).x by TAYLOR_1:def 5
    .=((sin `| Z)`|Z).x by A3,FDIFF_2:16
    .=((cos | Z)`|Z).x by TAYLOR_2:17
    .=(cos `| Z).x by A2,FDIFF_2:16
    .=diff(cos,x) by A1,A2,FDIFF_1:def 7
    .=-sin.x by SIN_COS:63;
  hence thesis;
end;
