theorem Th23:
  for G being Group, A being Subset of G holds
  (for g being Element of G st g in A holds g" in A) implies A" = A
proof
  let G be Group, A be Subset of G;
  assume
A1: for g being Element of G st g in A holds g" in A;
  thus A" c= A
  proof
    let x be object;
    assume x in A";
    then ex g being Element of G st x = g" & g in A;
    hence thesis by A1;
  end;
  let x be object;
  assume
A2: x in A;
  then reconsider a = x as Element of G;
A3: x = a"";
  a" in A by A1,A2;
  hence thesis by A3;
end;
