reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;

theorem
  for R being add-associative right_zeroed Abelian right_complementable
non empty addLoopStr for x,y being Element of R ex z being Element of R st x
  = y+z & x = z+y
proof
  let R be add-associative right_zeroed Abelian right_complementable non
  empty addLoopStr;
  let x,y be Element of R;
  take z = -y+x;
  z+y = x+(-y+y) by RLVECT_1:def 3
    .= x+0.R by RLVECT_1:5
    .= x by RLVECT_1:4;
  hence thesis;
end;
