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 Th9:
  (inverse_op G).:A = A"
proof
  set f = inverse_op G;
  hereby
    let y be object;
    assume y in f.:A;
    then consider x being object such that
A1: x in the carrier of G and
A2: x in A and
A3: y = f.x by FUNCT_2:64;
    reconsider x as Element of G by A1;
    y = x" by A3,GROUP_1:def 6;
    hence y in A" by A2;
  end;
  let y be object;
  assume y in A";
  then consider g being Element of G such that
A4: y = g" & g in A;
  f.g = g" by GROUP_1:def 6;
  hence thesis by A4,FUNCT_2:35;
end;
