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).3).x = (-cos).x
proof
  assume x in Z;
  then
A1: x in dom (cos|Z) by TAYLOR_2:17;
  dom ((-cos)|Z) = dom (-cos) /\ Z by RELAT_1:61
    .= dom cos /\ Z by VALUED_1:8
    .= dom (cos|Z) by RELAT_1:61
    .= dom (-(cos|Z)) by VALUED_1:8;
  then
A2: x in dom ((-cos)|Z) by A1,VALUED_1:8;
  (diff(sin,Z).3).x = (diff(sin,Z).(2*1+1)).x
    .= (((-1) |^ 1) (#) (cos | Z)).x by TAYLOR_2:19
    .=((-1)(#)(cos | Z)).x
    .=((-cos)|Z).x by RFUNCT_1:49
    .=(-cos).x by A2,FUNCT_1:47;
  hence thesis;
end;
