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 RLVECT_1:def 3
      .= 0_G + b by Def5
      .= x by Def4;
    -a in H by A1,Th51;
    hence thesis by A5,A6,Th50,Th103;
  end;
  assume
A7: a + H = carr(H);
  a + 0_G = a & 0_G in H by Th46,Def4;
  hence thesis by A7,Th103;
end;
