let r, s be Real; :: thesis: for n being Nat
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 & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let n be Nat; :: 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 & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
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 & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
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 & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let G be IntervalCoverPts of C; :: thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G implies G . (n + 1) < upper_bound (C /. n) )
assume ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G ) ; :: thesis: G . (n + 1) < upper_bound (C /. n)
then G . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ by Def3;
hence G . (n + 1) < upper_bound (C /. n) by XXREAL_1:4; :: thesis: verum