 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;

theorem
  {g}" = {g"}
proof
  thus {g}" c= {g"}
  proof
    let x be object;
    assume x in {g}";
    then consider h such that
A1: x = h" and
A2: h in {g};
    h = g by A2,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x be object;
  assume x in {g"};
  then
A3: x = g" by TARSKI:def 1;
  g in {g} by TARSKI:def 1;
  hence thesis by A3;
end;
