theorem
  inverse_op(H) = inverse_op(G) | the carrier of H
proof
  the carrier of H c= the carrier of G by Def5;
  then
A1: (the carrier of G) /\ (the carrier of H) = the carrier of H by XBOOLE_1:28;
A2: now
    let x be object;
    assume x in dom(inverse_op(H));
    then reconsider a = x as Element of H;
    reconsider b = a as Element of G by Th42;
    thus inverse_op(H).x = a" by GROUP_1:def 6
      .= b" by Th48
      .= inverse_op(G).x by GROUP_1:def 6;
  end;
  dom(inverse_op(H)) = the carrier of H & dom(inverse_op(G)) = the carrier
  of G by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:46;
end;
