theorem Th3:
for a,b,p be Real
   st 0 < p & 0 <= a & a < b holds a to_power p < b to_power p
proof
   let a,b,p be Real;
   assume A1: 0 < p & 0 <= a & a< b;
   now assume a = 0; then
    a to_power p = 0 by A1,POWER:def 2;
    hence a to_power p < b to_power p by A1,POWER:34;
   end;
   hence thesis by A1,POWER:37;
 end;
