let r, s be Real; for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let n be Nat; for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let C be IntervalCover of F; ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C implies not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty )
assume
( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C )
; not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
then
lower_bound (C /. (n + 1)) < upper_bound (C /. n)
by Def2;
hence
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
by XXREAL_1:33; verum