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 Th31:
  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_1:139;
 (a / b) #R c = a #R c / b #R c by A1,A2,PREPOWER:80;
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;
