
theorem Th5:
  for F being commutative associative well-unital
  almost_left_invertible right_zeroed non empty doubleLoopStr holds
  F is Euclidian
proof
  let F be commutative associative well-unital almost_left_invertible
  right_zeroed non empty doubleLoopStr;
  set f = the Function of F,NAT;
  for a,b being Element of F st b <> 0.F ex q,r being Element of F
  st a = q * b + r & (r = 0.F or f.r < f.b) by Lm4;
  hence thesis;
end;
