
theorem Th13:
  for X being set, x being Element of X, L being add-associative
right_zeroed right_complementable well-unital right-distributive
  non trivial doubleLoopStr holds Support 1_1(x,L) = {UnitBag x}
proof
  let X be set, x be Element of X, L be add-associative right_zeroed
  right_complementable well-unital right-distributive non trivial
  doubleLoopStr;
  now
    let a be object;
    hereby
      assume
A1:   a in Support 1_1(x,L);
      then reconsider b = a as Element of Bags X;
      now
        assume a <> UnitBag x;
        then 1_1(x,L).b = (0_(X,L)).b by FUNCT_7:32
          .= 0.L by POLYNOM1:22;
        hence contradiction by A1,POLYNOM1:def 4;
      end;
      hence a in {UnitBag x} by TARSKI:def 1;
    end;
    assume
A2: a in {UnitBag x};
    then a = UnitBag x by TARSKI:def 1;
    then 1_1(x,L).a <> 0.L by Th12;
    hence a in Support 1_1(x,L) by A2,POLYNOM1:def 4;
  end;
  hence thesis by TARSKI:2;
end;
