reserve a, b, r, s for Real;
reserve n, m for Nat,
  F for Subset-Family of Closed-Interval-TSpace (r,s);

theorem
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected & r <=
  s & [.r,s.] in F implies <*[.r,s.]*> is IntervalCover of F
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r <= s & [.r,s.] in F;
  set f = <*[.r,s.]*>;
A3: for n being Nat st 1 <= n holds (n <= len f implies f/.n is
non empty) & (n+1 <= len f implies lower_bound(f/.n) <= lower_bound(f/.(n+1)) &
  upper_bound(f/.n) <= upper_bound(f/.(n+1)) & lower_bound(f/.(n+1)) <
upper_bound(f/.n)) & (n+2 <= len f implies upper_bound(f/.n) <= lower_bound(f/.
  (n+2))) by A2,Lm3;
  rng f c= F & union rng f = [.r,s.] by A2,Lm3;
  hence thesis by A1,A2,A3,Def2;
end;
