
theorem Th4:
  for F be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, x,y being Element of F holds
  x*(-y) = -x*y
proof
  let F be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, x,y be Element of F;
  x*y +x*(-y) = x*(y+(-y)) by Def2
    .= x*(0.F) by RLVECT_1:def 10
    .= 0.F;
  hence thesis by RLVECT_1:def 10;
end;
