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 Th63:
  r3 in Cl X iff for O being open Subset of REAL st r3 in O holds
  O /\ X is non empty
proof
  hereby
    assume
A1: r3 in Cl X;
    let O be open Subset of REAL such that
A2: r3 in O and
A3: O /\ X is empty;
    O misses X by A3; then
A4: X c= O` by SUBSET_1:23;
A5: O misses O` by SUBSET_1:24;
    O` is closed by RCOMP_1:def 5;
    then Cl X c= O` by A4,Th57;
    hence contradiction by A1,A2,A5,XBOOLE_0:3;
  end;
A6: (Cl X)` is open;
  X c= Cl X & (Cl X)` /\ X c= X by Th58,XBOOLE_1:17;
  then
A7: (Cl X)` /\ X c= Cl X;
  (Cl X)` /\ X c= (Cl X)` by XBOOLE_1:17; then
A8: (Cl X)` /\ X is empty by A7,SUBSET_1:20;
   reconsider rr3=r3 as Element of REAL by XREAL_0:def 1;
  assume for O being open Subset of REAL st r3 in O holds O /\ X is non empty;
  then not rr3 in (Cl X)` by A6,A8;
  hence thesis by SUBSET_1:29;
end;
