
theorem
  for n,m being Nat, r being Real st
    m is odd & n >= m & r < 0 & r > -1 holds
  r to_power n >= r to_power m
  proof
    let n,m be Nat;
    let r be Real;
    assume A1: m is odd;
    assume A2: n >= m;
    assume A3: r < 0 & r > -1;
A4: n + 1 > m + 0 by A2,XREAL_1:8;
    per cases by A4,NAT_1:22;
    suppose n = m;
      hence thesis;
    end;
    suppose A5: n > m;
    then reconsider t = n-m as Nat by NAT_1:21;
A6: r to_power n - r to_power m = r to_power (t + m) - r to_power m
    .= r to_power t * r to_power m - 1 * r to_power m by Th2,A3
    .= (r to_power t - 1) * r to_power m;
A7: r to_power m <= 0 by Th7,A3,A1;
A8: t <> 0 by A5;
A9:  |.r.| <= 1 by A3,ABSVALUE:5;
    |.r.| <> 1
    proof
      assume |.r.| = 1; then
      |.r.| = |.1 .|;
      hence thesis by A3,ABSVALUE:28;
    end; then
A10: |.r.| < 1 by A9,XXREAL_0:1;
    |.r.| > 0 & t > 0 by A8,A3; then
    (|.r.|) to_power t < 1 to_power t by A10,POWER:37; then
    (|.r.|) to_power t < 1; then
    |.r to_power t.| < 1 & r to_power t <=
    |.r to_power t.| by ABSVALUE:4,SERIES_1:2; then
    r to_power t < 1 by XXREAL_0:2; then
    r to_power t - 1 <= 1 - 1 by XREAL_1:9; then
    r to_power n - r to_power m + r to_power m >=
    0 + r to_power m by A6,A7,XREAL_1:6;
    hence thesis;
    end;
  end;
