reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

theorem
  for x,y,z being Element of X st x in singleton X.z & y in singleton X.
  z holds x = y
proof
  let x,y,z be Element of X;
  assume that
A1: x in singleton X.z and
A2: y in singleton X.z;
  x = z by A1,Th52;
  hence thesis by A2,Th52;
end;
