let r, s be Real; :: thesis: 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 holds
1 <= len C
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); :: thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
1 <= len C
let C be IntervalCover of F; :: thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies 1 <= len C )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; :: thesis: 1 <= len C
assume not 1 <= len C ; :: thesis: contradiction
then (len C) + 1 <= 0 + 1 by NAT_1:13;
then A3: C is empty by XREAL_1:6;
union (rng C) = [.r,s.] by A1, A2, Def2;
hence contradiction by A2, A3, RELAT_1:38, XXREAL_1:1, ZFMISC_1:2; :: thesis: verum