reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th12:
  rng inverse_op G = the carrier of G
proof
  set f = inverse_op G;
  thus rng f c= the carrier of G;
  let x be object;
A1: dom f = the carrier of G by FUNCT_2:def 1;
  assume x in the carrier of G;
  then reconsider a = x as Element of G;
  f.(a") = a"" by GROUP_1:def 6
    .= a;
  hence thesis by A1,FUNCT_1:def 3;
end;
