reserve Z for open Subset of REAL;

theorem
  for r, e be Real st 0 < r & 0 < e
  ex n be Nat st
  for m be Nat st n <= m for x being Real st x in ].-r,r.[
  holds |.sin.x-Partial_Sums(Maclaurin(sin,].-r,r.[,x)).m.| < e & |.cos.x-
  Partial_Sums(Maclaurin(cos,].-r,r.[,x)).m.| < e
proof
  let r, e be Real such that
A1: r > 0 and
A2: e > 0;
  consider n be Nat such that
A3: for m be Nat st n <= m
  for x,s be Real st x in ].-r
,r.[ & 0 < s & s < 1 holds |.(diff(sin,].-r,r.[).m).(s*x) * x |^ m /(m!).|< e
  & |.(diff(cos,].-r,r.[).m).(s*x) * x |^ m /(m!).|< e by A1,A2,Th23;
  take n;
  let m be Nat such that
A4: n <= m;
A5: cos is_differentiable_on m+1, ].-r,r.[ & dom cos =REAL by Th21,
FUNCT_2:def 1;
A6: sin is_differentiable_on m+1, ].-r,r.[ & dom sin = REAL by Th21,
FUNCT_2:def 1;
  now
    m <= m+1 by NAT_1:11;
    then
A7: n <= m+1 by A4,XXREAL_0:2;
    let x be Real such that
A8: x in ].-r,r.[;
    ex s be Real
     st 0 < s & s < 1 & |.sin.x-Partial_Sums( Maclaurin(sin,
].-r,r.[,x)).m.| =|.(diff(sin,].-r,r.[).(m+1)).(s*x) * x |^ (m+1 ) / ((m+1)!).|
    by A1,A6,A8,Th4;
    hence |.sin.x-Partial_Sums(Maclaurin(sin,].-r,r.[,x)).m.| < e by A3,A8,A7;
    ex s be Real
       st 0 < s & s < 1 & |.cos.x-Partial_Sums( Maclaurin(cos,
].-r,r.[,x)).m.| =|.(diff(cos,].-r,r.[).(m+1)).(s*x) * x |^ (m+1 ) / ((m+1)!).|
    by A1,A5,A8,Th4;
    hence |.cos.x-Partial_Sums(Maclaurin(cos,].-r,r.[,x)).m.| < e by A3,A8,A7;
  end;
  hence thesis;
end;
