reserve a, b, p, q for Real;

theorem Th3:
  for p be Real st 0 <= p
  for a,b be Real st 0 <= a & a <= b holds
  a to_power p <= b to_power p
proof
  let p be Real such that
A1: 0 <= p;
  let a, b be Real such that
A2: 0 <= a and
A3: a <= b;
  per cases by A1;
  suppose
S1: 0 = p;
    a to_power 0 = 1 & b to_power 0 = 1 by POWER:24;
    hence thesis by S1;
  end;
  suppose
S2: 0 < p;
  per cases;
  suppose a = b;
    hence thesis;
  end;
  suppose
A4: a <> b;
A5: a < b by A3,A4,XXREAL_0:1;
    now
      per cases;
      suppose
A6:     a=0;
        then a to_power p = 0 by S2,POWER:def 2;
        hence thesis by A3,A4,A6,POWER:34;
      end;
      suppose a <> 0;
        hence thesis by A2,A5,S2,POWER:37;
      end;
    end;
    hence thesis;
  end;
end;
end;
