reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;

theorem Th26:
  the_set_of_all_left_open_real_bounded_intervals is
     cap-closed diff-finite-partition-closed with_empty_element
     with_countable_Cover
  proof
    the_set_of_all_left_open_real_bounded_intervals =
      {].a,b.] where a,b is Real:a <= b}
    proof
      hereby
        let x be object;
        assume x in the_set_of_all_left_open_real_bounded_intervals;
        then consider a,b be Real such that
A1:     x = ].a,b.];
        per cases;
        suppose a <= b;
          hence x in {].a,b.] where a,b is Real:a <= b} by A1;
        end;
        suppose not a <= b;
          then x = ]. 0 , 0 .] by A1,XXREAL_1:26;
          hence x in {].a,b.] where a,b is Real:a <= b};
        end;
      end;
      let x be object;
      assume x in {].a,b.] where a,b is Real:a <= b};
      then ex a,b be Real st x = ].a,b.] & a <= b;
      hence x in the_set_of_all_left_open_real_bounded_intervals;
    end;
    hence thesis by SRINGS_2:9;
  end;
