reserve T for TopSpace,
  x, y, a, b, U, Ux, rx for set,
  p, q for Rational,
  F, G for Subset-Family of T,
  Us, I for Subset-Family of Sorgenfrey-line;

theorem Th1:
  for x, a, b being Real st x in ].a,b.[
    ex p,r being Rational st x in ].p,r.[ & ].p,r.[ c= ].a,b.[
proof
   let x,a,b be Real such that
 A1: x in ].a,b.[;
 A2: x > a & x < b by XXREAL_1:4, A1;
   consider p being Rational such that A3: p > a & x > p by A2, RAT_1:7;
   consider r being Rational such that A4: x < r & r < b by A2, RAT_1:7;
   take p,r;
   thus x in ].p, r.[ by A3, A4, XXREAL_1:4;
   thus ].p, r.[ c= ].a,b.[
   proof
     let y be object; assume
  A5: y in ].p, r.[; then
     reconsider y1 = y as Element of REAL;
     y1 > p & y1 < r by XXREAL_1:4, A5; then
     y1 > a & y1 < b by A3, A4, XXREAL_0:2;
     hence y in ].a, b.[ by XXREAL_1:4;
   end;
 end;
