reserve Z for open Subset of REAL;

theorem Th26:
  for r, x be Real, m being Nat st 0 < r & m > 0 holds
  Partial_Sums(Maclaurin(sin,].-r,r.[,x)).(2*m) = Partial_Sums(x P_sin).(m-1) &
  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 and
A2: m > 0;
A3: m-1 is Element of NAT by A2,NAT_1:20;
  2*m > 2*0 by A2,XREAL_1:68;
  then
A4: 2*m-1 is Element of NAT by NAT_1:20;
  then
A5: Partial_Sums(Maclaurin(sin,].-r,r.[,x)).(2*m) = Partial_Sums(Maclaurin(
  sin,].-r,r.[,x)).(2*m-1) + Maclaurin(sin,].-r,r.[,x).((2*m-1)+1) by
SERIES_1:def 1
    .= Partial_Sums(Maclaurin(sin,].-r,r.[,x)).(2*m-1) + 0 by A1,Th20
    .= Partial_Sums(Maclaurin(sin,].-r,r.[,x)).(2*(m-1)+1)
    .= Partial_Sums(x P_sin).(m-1) by A1,A3,Th25;
  Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*m) = Partial_Sums(Maclaurin(
cos,].-r,r.[,x)).(2*m-1) + Maclaurin(cos,].-r,r.[,x).((2*m-1)+1) by A4,
SERIES_1:def 1
    .= Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*m-1) + (-1) |^ m * x |^ (2
  *m) / ((2*m)!) by A1,Th20
    .= Partial_Sums(Maclaurin(cos,].-r,r.[,x)).(2*(m-1)+1) + (x P_cos).m by
SIN_COS:def 21
    .= Partial_Sums(x P_cos).((m-1)) + (x P_cos).((m-1)+1) by A1,A3,Th25
    .= Partial_Sums(x P_cos).m by A3,SERIES_1:def 1;
  hence thesis by A5;
end;
