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 GROUP_1:def 3
      .= b * 1_G by GROUP_1:def 5
      .= x by GROUP_1:def 4;
    a" in H by A1,Th51;
    then b * a" in H by A5,Th50;
    hence thesis by A6,Th104;
  end;
  assume
A7: H * a = carr(H);
  1_G * a = a & 1_G in H by Th46,GROUP_1:def 4;
  then a in carr(H) by A7,Th104;
  hence thesis;
end;
