theorem
  {g} + {g1,g2} = {g + g1,g + g2}
proof
  thus {g} + {g1,g2} c= {g + g1,g + g2}
  proof
    let x be object;
    assume x in {g} + {g1,g2};
    then consider h1,h2 such that
A1: x = h1 + h2 and
A2: h1 in {g} and
A3: h2 in {g1,g2};
A4: h2 = g1 or h2 = g2 by A3,TARSKI:def 2;
    h1 = g by A2,TARSKI:def 1;
    hence thesis by A1,A4,TARSKI:def 2;
  end;
  let x be object;
A5: g2 in {g1,g2} by TARSKI:def 2;
  assume x in {g + g1,g + g2};
  then
A6: x = g + g1 or x = g + g2 by TARSKI:def 2;
  g in {g} & g1 in {g1,g2} by TARSKI:def 1,def 2;
  hence thesis by A6,A5;
end;
