
theorem
  for R being well-unital associative non empty doubleLoopStr, I being
  non empty Subset of R, a being Element of R holds a in sqrt I iff ex n being
  Element of NAT st a|^n in sqrt I
proof
  let R be well-unital associative non empty doubleLoopStr, I be non empty
  Subset of R, a be Element of R;
A1: now
    assume ex n being Element of NAT st a|^n in sqrt I;
    then consider n being Element of NAT such that
A2: a|^n in sqrt I;
    consider d being Element of R such that
A3: a|^n = d and
A4: ex m being Element of NAT st d|^m in I by A2;
    consider m being Element of NAT such that
A5: d|^m in I by A4;
    a|^(n*m) = d|^m by A3,BINOM:11;
    hence a in sqrt I by A5;
  end;
  now
A6: a|^1 = a by BINOM:8;
    assume a in sqrt I;
    hence ex n being Element of NAT st a|^n in sqrt I by A6;
  end;
  hence thesis by A1;
end;
