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(cos,Z).3).x = sin.x
proof
  assume x in Z;
  then
A1: x in dom (sin|Z) by TAYLOR_2:17;
  (diff(cos,Z).3).x = (diff(cos,Z).(2*1+1)).x
    .= (((-1) |^ (1+1)) (#) (sin | Z)).x by TAYLOR_2:19
    .=(((1|^2)(#)sin| Z)).x by WSIERP_1:1
    .=(((1*1)(#)sin| Z)).x
    .=((sin)|Z).x by RFUNCT_1:21
    .=(sin).x by A1,FUNCT_1:47;
  hence thesis;
end;
