theorem
  add_inverse(H) = add_inverse(G) | the carrier of H
proof
A1: (the carrier of G) /\ (the carrier of H) = the carrier of H
    by DefA5,XBOOLE_1:28;
A2: now
    let x be object;
    assume x in dom(add_inverse(H));
    then reconsider a = x as Element of H;
    reconsider b = a as Element of G by Th41,STRUCT_0:def 5;
    thus add_inverse(H).x = -a by Def6
      .= -b by Th48
      .= add_inverse(G).x by Def6;
  end;
  dom(add_inverse(H)) = the carrier of H & dom(add_inverse(G)) = the carrier
  of G by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:46;
end;
