reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem
  a>0 & b>0 implies (a*b) to_power c = a to_power c*b to_power c
proof
  assume that
A1: a > 0 and
A2: b > 0;
A3: a * b > 0 by A1,A2,XREAL_1:129;
 (a * b) #R c = a #R c * b #R c by A1,A2,PREPOWER:78;
then  (a * b) #R c = a #R c * b to_power c by A2,Def2;
then  (a * b) #R c = a to_power c * b to_power c by A1,Def2;
  hence thesis by A3,Def2;
end;
