let a, c be Real; :: thesis: ( a > 0 implies a to_power (- c) = 1 / (a to_power c) )
assume A1: a > 0 ; :: thesis: a to_power (- c) = 1 / (a to_power c)
then a #R (- c) = 1 / (a #R c) by PREPOWER:76;
then a #R (- c) = 1 / (a to_power c) by A1, Def2;
hence a to_power (- c) = 1 / (a to_power c) by A1, Def2; :: thesis: verum