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
  r>0 implies Maclaurin(sin,].-r,r.[,x).n = sin.(n*PI/2)*x|^ n / (n!) &
  Maclaurin(cos,].-r,r.[,x).n = cos.(n*PI/2)*x|^ n / (n!)
proof
A1: |.0-0.|=0 by ABSVALUE:2;
  assume r>0;
  then
A2: 0 in ].0-r,0+r.[ by A1,RCOMP_1:1;
A3: Maclaurin(cos,].-r,r.[,x).n =Taylor(cos,].-r,r.[,0,x).n by TAYLOR_2:def 1
    .=(diff(cos,].-r,r.[).n).0 *(x-0)|^ n / (n!) by TAYLOR_1:def 7
    .=cos.(0+n*PI/2)*x|^ n / (n!) by A2,Th14;
  Maclaurin(sin,].-r,r.[,x).n =Taylor(sin,].-r,r.[,0,x).n by TAYLOR_2:def 1
    .=(diff(sin,].-r,r.[).n).0 *(x-0)|^ n / (n!) by TAYLOR_1:def 7
    .=sin.(0+n*PI/2)*x|^ n / (n!) by A2,Th11;
  hence thesis by A3;
end;
