
theorem
  for A being Subset of REAL, B being Subset of R^1 st A = B holds Cl A = Cl B
proof
  let A be Subset of REAL, B be Subset of R^1 such that
A1: A = B;
  thus Cl A c= Cl B
  proof
    let a be object;
    assume
A2: a in Cl A;
    for G being Subset of R^1 st G is open & a in G holds B meets G
    proof
      let G be Subset of R^1 such that
A3:   G is open and
A4:   a in G;
      reconsider H = G as Subset of REAL by TOPMETR:17;
      G in Family_open_set RealSpace by A3,Lm8;
      then H is open by Th5;
      then A /\ H is non empty by A2,A4,MEASURE6:63;
      hence thesis by A1;
    end;
    hence thesis by A2,PRE_TOPC:def 7,TOPMETR:17;
  end;
  let a be object;
  assume
A5: a in Cl B;
  for O being open Subset of REAL st a in O holds O /\ A is non empty
  proof
    let O be open Subset of REAL such that
A6: a in O;
    reconsider P = O as Subset of R^1 by TOPMETR:17;
    P in Family_open_set RealSpace by Th5;
    then P is open by Lm8;
    then P meets B by A5,A6,PRE_TOPC:def 7;
    hence thesis by A1;
  end;
  hence thesis by A5,MEASURE6:63;
end;
