let a, b, c be Real; :: thesis: ( a < b & c > 1 implies c to_power a < c to_power b )
assume that
A1: a < b and
A2: c > 1 ; :: thesis: c to_power a < c to_power b
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 c to_power a < c to_power b by A4, XCMPLX_1:87; :: thesis: verum