reserve Z for open Subset of REAL;

theorem Th27:
  for r, x be Real, m be Nat st 0 < r holds
  Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*m) = Partial_Sums(x P_cos).m
proof
  let r, x be Real, m be Nat such that
A1: r > 0;
  defpred P[Nat] means Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*
  $1) = Partial_Sums(x P_cos).$1;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
    P[k];
    thus Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*(k+1)) = Partial_Sums(
Maclaurin(cos,].-r,r.[,x)).(2*k+1) + Maclaurin(cos,].-r,r.[,x).((2*k+1)+1) by
SERIES_1:def 1
      .= Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*k+1) + (-1) |^ (k+1) * x
    |^ (2*(k+1)) / ((2*(k+1))!) by A1,Th20
      .= Partial_Sums(x P_cos).k + (-1) |^ (k+1) * x |^ (2*(k+1)) / ((2*(k+1
    ))!) by A1,Th25
      .= Partial_Sums(x P_cos).k + (x P_cos).(k+1) by SIN_COS:def 21
      .= Partial_Sums(x P_cos).(k+1) by SERIES_1:def 1;
  end;
  Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*0) = Maclaurin(cos,].-r,r.[,x
  ).(2*0) by SERIES_1:def 1
    .= (-1) |^ 0 * x |^ (2*0) / ((2*0)!) by A1,Th20
    .= (x P_cos).0 by SIN_COS:def 21
    .= Partial_Sums(x P_cos).0 by SERIES_1:def 1;
  then
A3: P[0];
  for n be Nat holds P[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
