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;
reserve n for Nat,
        S for Subset-Family of REAL;

theorem
  for n being non zero Nat,
    x being Element of Product(n,the_set_of_all_closed_real_bounded_intervals)
  holds (ex a,b being Element of REAL n st for t being Element of REAL n holds
    (t in x iff (for i being Nat st i in Seg n holds t.i in [. a.i, b.i .])))
  proof
    let n being non zero Nat,
      x be Element of Product(n,the_set_of_all_closed_real_bounded_intervals);
    consider s being Tuple of n,the_set_of_all_closed_real_bounded_intervals
      such that
A1: for t being Element of REAL n holds
     (for i being Nat st i in Seg n holds t.i in s.i) iff t in x by Th42;
    consider a,b being Element of REAL n such that
A2: for i being Nat st i in Seg n holds s.i = [.a.i,b.i.] by Th43;
    take a,b;
      let t be Element of REAL n;
      hereby
        assume
A3:     t in x;
        thus for i be Nat st i in Seg n holds t.i in [.a.i,b.i.]
        proof
          let i be Nat;
          assume
A4:       i in Seg n;
          then s.i = [.a.i,b.i.] by A2;
          hence t.i in [.a.i,b.i.] by A1,A3,A4;
        end;
      end;
      assume
A5:   for i be Nat st i in Seg n holds t.i in [.a.i,b.i.];
      for i be Nat st i in Seg n holds t.i in s.i
      proof
        let i be Nat;
        assume
A6:     i in Seg n;
        then s.i = [.a.i,b.i.] by A2;
        hence t.i in s.i by A6,A5;
      end;
      hence t in x by A1;
  end;
