let a, b, c be Real; :: thesis: ( a > 0 implies (a to_power b) to_power c = a to_power (b * c) )
assume A1: a > 0 ; :: thesis: (a to_power b) to_power c = a to_power (b * c)
then A2: a #R b > 0 by PREPOWER:81;
(a #R b) #R c = a #R (b * c) by A1, PREPOWER:91;
then (a #R b) #R c = a to_power (b * c) by A1, Def2;
then (a #R b) to_power c = a to_power (b * c) by A2, Def2;
hence (a to_power b) to_power c = a to_power (b * c) by A1, Def2; :: thesis: verum