
theorem Th45:
  for L being add-associative right_zeroed right_complementable
  left-distributive well-unital non empty doubleLoopStr, x being Element of L
  holds Roots <%-x, 1.L%> = {x}
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  well-unital non empty doubleLoopStr, x be Element of L;
  now
    let a be object;
    hereby
      assume
A1:   a in Roots <%-x, 1.L%>;
      then reconsider b = a as Element of L;
      b is_a_root_of <%-x, 1.L%> by A1,POLYNOM5:def 10;
      then 0.L = eval(<%-x, 1.L%>,b)
        .= -x + b by POLYNOM5:47;
      then -x = -b by RLVECT_1:6;
      hence x = a by RLVECT_1:18;
    end;
    eval(<%-x, 1.L%>,x) = -x + x by POLYNOM5:47
      .= 0.L by RLVECT_1:5;
    then
A2: x is_a_root_of <%-x, 1.L%>;
    assume a = x;
    hence a in Roots <%-x, 1.L%> by A2,POLYNOM5:def 10;
  end;
  hence thesis by TARSKI:def 1;
end;
