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 Th59:
  X is closed iff X = Cl X
proof
  hereby
    set ClX = { A where A is Subset of REAL : X c= A & A is closed };
    assume X is closed;
    then X in ClX; then
A1: Cl X c= X by SETFAM_1:3;
    X c= Cl X by Th58;
    hence X = Cl X by A1;
  end;
  thus thesis;
end;
