let a, b, c be Real; :: thesis: ( a > 0 implies a to_power (b + c) = (a to_power b) * (a to_power c) )
assume A1: a > 0 ; :: thesis: a to_power (b + c) = (a to_power b) * (a to_power c)
then a #R (b + c) = (a #R b) * (a #R c) by PREPOWER:75;
then a #R (b + c) = (a #R b) * (a to_power c) by A1, Def2;
then a #R (b + c) = (a to_power b) * (a to_power c) by A1, Def2;
hence a to_power (b + c) = (a to_power b) * (a to_power c) by A1, Def2; :: thesis: verum