let r, s be Real; :: 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 & len C = 1 holds
G = <*r,s*>
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 & len C = 1 holds
G = <*r,s*>
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 & len C = 1 holds
G = <*r,s*>
let G be IntervalCoverPts of C; :: thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 implies G = <*r,s*> )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) and
A2: len C = 1 ; :: thesis: G = <*r,s*>
A3: G . 1 = r by A1, Def3;
A4: len G = (len C) + 1 by A1, Def3;
then G . 2 = s by A1, A2, Def3;
hence G = <*r,s*> by A2, A4, A3, FINSEQ_1:44; :: thesis: verum