
theorem Th17:
  for L being associative commutative well-unital distributive
almost_left_invertible non empty doubleLoopStr, x being Element of L, n being
  Element of NAT holds pow(x,n+1) = pow(x,n) * x & pow(x,n+1) = x * pow(x,n)
proof
  let L be associative commutative well-unital distributive
  almost_left_invertible non empty doubleLoopStr;
  let x be Element of L;
  let n be Element of NAT;
  pow(x,n+1) = x |^ (n+1) by Def2
    .= (x |^ n) * x by GROUP_1:def 7
    .= pow(x,n) * x by Def2;
  hence thesis;
end;
