 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.[,
  S be Subset-Family of Closed-Interval-TSpace(r,s)
  st r <= s &
  S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
  x is Element of [.r,s.] holds
  for IC being IntervalCover of S holds
  IC is FinSequence of bool REAL & rng IC c= S
  & union rng IC = [.r,s.] & (for n being Nat st 1 <= n holds (n <=
  len IC implies IC/.n is non empty) & (n+1 <= len IC implies
    lower_bound(IC/.n)
  <= lower_bound(IC/.(n+1)) & upper_bound(IC/.n) <= upper_bound(IC/.(n+1)) &
  lower_bound(IC/.(n+1)) < upper_bound(IC/.n)) & (n+2 <= len IC implies
  upper_bound(IC/.n) <= lower_bound(IC/.(n+2)))) & ([.r,s.] in S implies
    IC = <*[.r,s.]*>) & (not [.r,s.] in S implies
    (ex p being Real st r < p & p <= s & IC.1 = [.r,p.[) &
    (ex p being Real st r <= p & p < s & IC.len IC = ].p,s.]) &
    for n being Nat st 1 < n & n < len IC ex p, q being Real st
    r <= p & p < q & q <= s & IC.n = ].p,q.[ )
  proof
    let jauge be Function of [.r,s.],].0,+infty.[,
    S be Subset-Family of Closed-Interval-TSpace(r,s);
    assume that
A1: r <= s and
A2: S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
    x is Element of [.r,s.];
    let IC be IntervalCover of S;
    S is Subset-Family of Closed-Interval-TSpace(r,s) &
    S is Cover of Closed-Interval-TSpace(r,s) &
    S is open connected & r <= s by A1,A2,Th39,Th40,Th41;
    hence thesis by RCOMP_3:def 2;
  end;
