 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;
reserve r,s for Real;

theorem
  for jauge being Function of [.r,s.],].0,+infty.[ st r <= s holds
  the set of all ]. x - jauge.x, x + jauge.x .[ /\ [.r,s.] where x is Element
  of [.r,s.] is Subset-Family of Closed-Interval-TSpace(r,s)
  proof
    let jauge be Function of [.r,s.],].0,+infty.[;
    assume
A1: r <= s;
    set A = the set of all ]. x - jauge.x, x + jauge.x .[ /\ [.r,s.] where
    x is Element of [.r,s.];
    A c= bool [.r,s.]
    proof
      let t be object;
      assume t in A;
      then consider x0 be Element of [.r,s.] such that
A2:   t = ]. x0 - jauge.x0,x0 + jauge.x0 .[ /\ [.r,s.];
      reconsider t as set by TARSKI:1;
      t c= [.r,s.] by A2,XBOOLE_1:17;
      hence thesis;
    end;
    hence thesis by A1,TOPMETR:18;
  end;
