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
  x0 in Z implies Taylor(sin,Z,x0,x).n = sin.(x0+n*PI/2)*(x-x0)|^ n / (n
  !) & Taylor(cos,Z,x0,x).n = cos.(x0+n*PI/2)*(x-x0)|^ n / (n!)
proof
  assume
A1: x0 in Z;
A2: Taylor(cos,Z,x0,x).n =(diff(cos,Z).n).x0 *(x-x0)|^ n / (n!) by
TAYLOR_1:def 7
    .=cos.(x0+n*PI/2)*(x-x0)|^ n / (n!) by A1,Th14;
  Taylor(sin,Z,x0,x).n =(diff(sin,Z).n).x0 *(x-x0)|^ n / (n!) by TAYLOR_1:def 7
    .=sin.(x0+n*PI/2)*(x-x0)|^ n / (n!) by A1,Th11;
  hence thesis by A2;
end;
