theorem Th113:
  a in H iff a * H = carr(H)
proof
  thus a in H implies a * H = carr(H)
  proof
    assume
A1: a in H;
    thus a * H c= carr(H)
    proof
      let x be object;
      assume x in a * H;
      then consider g such that
A2:   x = a * g and
A3:   g in H by Th103;
      a * g 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: a * (a" * b) = a * a" * b by GROUP_1:def 3
      .= 1_G * b by GROUP_1:def 5
      .= x by GROUP_1:def 4;
    a" in H by A1,Th51;
    then a" * b in H by A5,Th50;
    hence thesis by A6,Th103;
  end;
  assume
A7: a * H = carr(H);
  a * 1_G = a & 1_G in H by Th46,GROUP_1:def 4;
  then a in carr(H) by A7,Th103;
  hence thesis;
end;
