theorem Th37:
  0_G + A = A & A + 0_G = A
proof
  thus 0_G + A = A
  proof
    thus 0_G + A c= A
    proof
      let x be object;
      assume x in 0_G + A;
      then ex h st x = 0_G + h & h in A by Th27;
      hence thesis by Def4;
    end;
    let x be object;
    assume
A1: x in A;
    then reconsider a = x as Element of G;
    0_G + a = a by Def4;
    hence thesis by A1,Th27;
  end;
  thus A + 0_G c= A
  proof
    let x be object;
    assume x in A + 0_G;
    then ex h st x = h + 0_G & h in A by Th28;
    hence thesis by Def4;
  end;
  let x be object;
  assume
A2: x in A;
  then reconsider a = x as Element of G;
  a + 0_G = a by Def4;
  hence thesis by A2,Th28;
end;
