
theorem Th13:
  for L being add-associative right_zeroed right_complementable
  left-distributive left_unital non empty doubleLoopStr, I being left-ideal
  non empty Subset of L, x being Element of L st x in I holds -x in I
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  left_unital non empty doubleLoopStr;
  let I be left-ideal non empty Subset of L;
  let x being Element of L;
  assume x in I;
  then
A1: (- 1.L)*x in I by Def2;
  0. L = 0.L*x
    .= (1.L + (- 1.L))*x by RLVECT_1:def 10
    .= 1.L*x + (- 1.L)*x by VECTSP_1:def 3
    .= x + (- 1.L)*x;
  hence thesis by A1,RLVECT_1:def 10;
end;
