reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem
  for X, Y being real-membered set holds
  X c= Y implies Cl X c= Cl Y
proof
  let X, Y be real-membered set;
  assume
A1: X c= Y;
  set ClX = { A where A is Subset of REAL : X c= A & A is closed };
  Y c= Cl Y by Th58;
  then X c= Cl Y by A1;
  then Cl Y in ClX;
  hence thesis by SETFAM_1:3;
end;
