let r, s be real number ; :: thesis: for F being Subset-Family of (Closed-Interval-TSpace r,s)
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace r,s) & F is open & F is connected & r <= s holds
2 <= len G
let F be Subset-Family of (Closed-Interval-TSpace r,s); :: thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace r,s) & F is open & F is connected & r <= s holds
2 <= len G
let C be IntervalCover of F; :: thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace r,s) & F is open & F is connected & r <= s holds
2 <= len G
let G be IntervalCoverPts of C; :: thesis: ( F is Cover of (Closed-Interval-TSpace r,s) & F is open & F is connected & r <= s implies 2 <= len G )
assume that
A1: F is Cover of (Closed-Interval-TSpace r,s) and
A2: F is open and
A3: F is connected and
A4: r <= s ; :: thesis: 2 <= len G
1 <= len C by A1, A2, A3, A4, Th65;
then 1 + 1 <= (len C) + 1 by XREAL_1:8;
hence 2 <= len G by A1, A2, A3, A4, Def3; :: thesis: verum