theorem Th1:
  for O be non empty Ordinal, i be Element of O,
    L be add-associative right_zeroed
         right_complementable well-unital distributive non trivial
         doubleLoopStr
  for x be Function of O, L holds
    eval(1_1(i,L),x) = x.i
proof
  let O be non empty Ordinal, i be Element of O,
      L be add-associative right_zeroed right_complementable
           well-unital distributive non trivial doubleLoopStr;
  let x be Function of O, L;
  set p = 1_1(i,L);
  Support p = {UnitBag i} by HILBASIS:13;
  then eval(p, x) = (p.UnitBag i)*eval(UnitBag i,x) by POLYNOM2:19
  .= 1_L *eval(UnitBag i,x) by HILBASIS:12
  .= eval(UnitBag i,x);
  hence thesis by HILBASIS:11;
end;
