
theorem Th2:
  for a, k, r being Nat, x being Real
  st 1 < x & 0 < k holds x |^ (a*k + r) = (x |^ a)*(x |^ (a*(k-'1)+r))
proof
  let a,k,r be Nat, x be Real such that
A1: 1 < x and
A2: 0 < k;
  set xNak = x |^ (a*k + r);
  0+1 <= k by A2,NAT_1:13;
  then k = k-'1+1 by XREAL_1:235;
  then xNak = (x #Z (a + (a*(k-'1)+r))) by PREPOWER:36;
  then xNak = (x #Z a)*(x #Z (a*(k-'1)+r)) by A1,PREPOWER:44;
  then xNak = (x |^ a)*(x #Z (a*(k-'1)+r)) by PREPOWER:36;
  hence thesis by PREPOWER:36;
end;
