reserve S for 1-sorted,
  i for Element of NAT,
  p for FinSequence,
  X for set;

theorem
  for X,Y being set holds X = Y iff for x being object holds X@x = Y@x
proof
  let X,Y be set;
  thus X = Y implies for x being object holds X@x = Y@x;
  thus (for x being object holds X@x = Y@x) implies X = Y
  proof
    assume
A1: for x being object holds X@x = Y@x;
    thus X c= Y
    proof
      let y be object;
      assume y in X;
      then X@y = 1.Z_2 by Def3;
      then Y@y = 1.Z_2 by A1;
      hence thesis by Def3;
    end;
    let y be object;
    assume y in Y;
    then Y@y = 1.Z_2 by Def3;
    then X@y = 1.Z_2 by A1;
    hence thesis by Def3;
  end;
end;
