
theorem Th8:
  for a, b, c being Real st a <= b & c >= 1 holds
  c to_power a <= c to_power b
proof
  let a, b, c be Real;
  assume a <= b;
  then
A1: a < b or a = b by XXREAL_0:1;
  assume c >= 1;
  then per cases by XXREAL_0:1;
  suppose c > 1;
    hence thesis by A1,POWER:39;
  end;
  suppose
A2: c = 1;
    1 to_power a = 1 & 1 to_power b = 1 by POWER:26;
    hence thesis by A2;
  end;
end;
