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;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;

theorem
  the_set_of_all_open_real_bounded_intervals c=
    {I where I is Subset of REAL : I is open_interval}
  proof
    let x be object;
    assume x in the_set_of_all_open_real_bounded_intervals;
    then consider a,b be Real such that
A2: x = ].a,b.[;
    reconsider x1 = x as Subset of REAL by A2;
    a is Element of ExtREAL & b is Element of ExtREAL
      by NUMBERS:31,XREAL_0:def 1;
    then x1 is open_interval by A2,MEASURE5:def 2;
    hence thesis;
  end;
