reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  n>=1 implies (1+1/(n+1))|^n<(1+1/n)|^(n+1)
proof
  assume
A1: n>=1;
  n+1>n+0 by XREAL_1:8;
  then 1/(n+1)<1/n by A1,XREAL_1:76;
  then 1/(n+1)+1<1/n+1 by XREAL_1:8;
  then
A2: (1+1/(n+1))|^n<(1+1/n)|^n by A1,PREPOWER:10;
A3: (1+1/n)|^n>0 by PREPOWER:6;
  (1+1/n)|^(n+1)-(1+1/n)|^n =(1+1/n)|^n*(1+1/n)-(1+1/n)|^n by NEWTON:6
    .=(1+1/n)|^n*(1/n);
  then (1+1/n)|^(n+1)-(1+1/n)|^n+(1+1/n)|^n>0+(1+1/n)|^n by A1,A3,XREAL_1:8;
  hence thesis by A2,XXREAL_0:2;
end;
