
theorem Th14:
  for L being add-associative right_zeroed right_complementable
right-distributive right_unital non empty doubleLoopStr, I being right-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
  right-distributive right_unital non empty doubleLoopStr;
  let I be right-ideal non empty Subset of L;
  let x being Element of L;
  assume x in I;
  then
A1: x*(- 1.L) in I by Def3;
  0. L = x*0.L
    .= x*(1.L + (- 1.L)) by RLVECT_1:def 10
    .= x*1.L + x*(- 1.L) by VECTSP_1:def 2
    .= x + x*(- 1.L);
  hence thesis by A1,RLVECT_1:def 10;
end;
