reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;

theorem
  X (+) {x} = X+x
proof
  thus X (+) {x} c= X+x
  proof
    let p be object;
    assume p in X (+) {x};
    then consider y,z being Point of T such that
A1: p=y+z and
A2: y in X and
A3: z in {x};
    {z} c= {x} by A3,ZFMISC_1:31;
    then p=y+x by A1,ZFMISC_1:18;
    hence thesis by A2;
  end;
  let p be object;
  assume p in X+x;
  then
A4: ex q being Point of T st p=q+x & q in X;
  x in {x} by TARSKI:def 1;
  hence thesis by A4;
end;
