reserve X, Y for non empty set;

theorem Th1:
  for R,S being Membership_Func of X st for x being Element of X
  holds R.x = S.x holds R = S
proof
  let R,S be Membership_Func of X;
  assume for x being Element of X holds R.x = S.x;
  then
A1: for x being Element of X st x in dom R holds R.x = S.x;
  dom R = X & dom S = X by FUNCT_2:def 1;
  hence thesis by A1,PARTFUN1:5;
end;
