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 Th10:
  (inverse_op G)"A = A"
proof
  set f = inverse_op G;
A1: dom f = the carrier of G by FUNCT_2:def 1;
  hereby
    let x be object;
    assume
A2: x in f"A;
    then reconsider g = x as Element of G;
    f.x in A by A2,FUNCT_1:def 7;
    then (f.g)" in A";
    then g"" in A" by GROUP_1:def 6;
    hence x in A";
  end;
  let x be object;
  assume x in A";
  then consider g being Element of G such that
A3: x = g" & g in A;
  f.(g") = g"" by GROUP_1:def 6
    .= g;
  hence thesis by A1,A3,FUNCT_1:def 7;
end;
