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 Th58:
  for X being real-membered set holds X c= Cl X
proof
  let X be real-membered set;
  let x be object;
  set ClX = { A where A is Subset of REAL : X c= A & A is closed };
  assume
A1: x in X;
A2: now
    let Y be set;
    assume Y in ClX;
    then ex YY being Subset of REAL st YY = Y & X c= YY & YY is closed;
    hence x in Y by A1;
  end;
  X c= [#]REAL by MEMBERED:3; then
  [#]REAL in ClX;
  hence thesis by A2,SETFAM_1:def 1;
end;
