let a, b, c be real number ; :: thesis: ( a < b & c > 1 implies c to_power a < c to_power b )
assume A1: ( a < b & c > 1 ) ; :: thesis: c to_power a < c to_power b
then A2: c to_power a > 0 by Th39;
A3: c to_power a <> 0 by A1, Th39;
b - a > 0 by A1, XREAL_1:52;
then c to_power (b - a) > 1 by A1, Th40;
then (c to_power b) / (c to_power a) > 1 by A1, Th34;
then ((c to_power b) / (c to_power a)) * (c to_power a) > 1 * (c to_power a) by A2, XREAL_1:70;
hence c to_power a < c to_power b by A3, XCMPLX_1:88; :: thesis: verum