
theorem Th73:
  for R being left_zeroed right_zeroed right_add-cancelable
right-distributive non empty doubleLoopStr, I,J being right-ideal non empty
  Subset of R holds I c= I + J
proof
  let R be left_zeroed right_add-cancelable right_zeroed right-distributive
  non empty doubleLoopStr, I,J be right-ideal non empty Subset of R;
    let u be object;
    assume u in I;
    then reconsider u9 = u as Element of I;
    0.R is Element of J by Th3;
    then u9 + 0.R in {a + b where a,b is Element of R : a in I & b in J};
    hence u in I + J by RLVECT_1:def 4;
end;
