reserve Z for open Subset of REAL;

theorem
  for n be Element of NAT, r,x,s be Real st x in ].-r,r.[ & 0 < s & s <
1 holds |.(diff(exp_R,].-r,r.[).(n+1)).(s*x) * x |^ (n+1) / ((n+1)!) .| <=
  |.exp_R.(s*x).| * |.x.| |^ (n+1) / ((n+1)!)
proof
  let n be Element of NAT;
  let r,x,s be Real such that
A1: x in ].-r,r.[ and
A2: 0 < s and
A3: s < 1;
A4: |.s*x-0.| = |.s.| * |.x.| by COMPLEX1:65
    .= s * |.x.| by A2,ABSVALUE:def 1;
  x in ].0-r,0+r.[ by A1;
  then
A5: |.x-0.| < r by RCOMP_1:1;
  |.x.|>=0 by COMPLEX1:46;
  then |.x.| * s < r * 1 by A2,A3,A5,XREAL_1:97;
  then (s*x) in ].0-r,0+r.[ by A4,RCOMP_1:1;
  then
A6: (s*x) in dom(exp_R | ].-r,r.[) by Th5;
A7: |.(n+1)!.| = ((n+1)!) by ABSVALUE:def 1;
  |. (diff(exp_R,].-r,r.[).(n+1)).(s*x) * x |^ (n+1) /((n+1)!).| = |.
  (exp_R | ].-r,r.[).(s*x) * x |^ (n+1) /((n+1)!).| by Th6
    .= |.exp_R.(s*x) * x |^ (n+1) /((n+1)!).| by A6,FUNCT_1:47
    .= |.exp_R.(s*x) * x |^ (n+1).| /|.(n+1)!.| by COMPLEX1:67
    .= |.exp_R.(s*x).| * |.x |^ (n+1).| /((n+1)!) by A7,COMPLEX1:65;
  hence thesis by Th1;
end;
