theorem Th119:
  a in H iff H + a = carr(H)
proof
  thus a in H implies H + a = carr(H)
  proof
    assume
A1: a in H;
    thus H + a c= carr(H)
    proof
      let x be object;
      assume x in H + a;
      then consider g such that
A2:   x = g + a and
A3:   g in H by Th104;
      g + a in H by A1,A3,Th50;
      hence thesis by A2;
    end;
    let x be object;
    assume
A4: x in carr(H);
    then
A5: x in H;
    reconsider b = x as Element of G by A4;
A6: (b + -a) + a = b + (-a + a) by RLVECT_1:def 3
      .= b + 0_G by Def5
      .= x by Def4;
    -a in H by A1,Th51;
    hence thesis by A5,A6,Th50,Th104;
  end;
  assume
A7: H + a = carr(H);
  0_G + a = a & 0_G in H by Th46,Def4;
  hence thesis by A7,Th104;
end;
