reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th39:
  a < b & c > 1 implies c to_power a < c to_power b
proof
  assume that
A1: a < b and
A2: c > 1;
A3: c to_power a > 0 by A2,Th34;
A4: c to_power a <> 0 by A2,Th34;
 b-a>0 by A1,XREAL_1:50;
then  c to_power (b-a) > 1 by A2,Th35;
then  c to_power b / c to_power a > 1 by A2,Th29;
then
 c to_power b/c to_power a*c to_power a > 1*c to_power a by A3,XREAL_1:68;
  hence thesis by A4,XCMPLX_1:87;
end;
