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
lower_bound (C /. 1) = r
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
lower_bound (C /. 1) = r
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 lower_bound (C /. 1) = r )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; :: thesis: lower_bound (C /. 1) = r
1 <= len C by A1, A2, Th51;
then A3: C . 1 = C /. 1 by FINSEQ_4:15;

per cases ( [.r,s.] in F or not [.r,s.] in F ) ;

suppose [.r,s.] in F ; :: thesis: lower_bound (C /. 1) = r

then C = <*[.r,s.]*> by A1, A2, Def2;
then C /. 1 = [.r,s.] by FINSEQ_4:16;
hence lower_bound (C /. 1) = r by A2, JORDAN5A:19; :: thesis: verum

end;

suppose not [.r,s.] in F ; :: thesis: lower_bound (C /. 1) = r

then ex p being Real st
( r < p & p <= s & C . 1 = [.r,p.[ ) by A1, A2, Def2;
hence lower_bound (C /. 1) = r by A3, Th4; :: thesis: verum

end;

end;