let r, s be real number ; :: 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
r in C /. 1
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
r in C /. 1
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 r in C /. 1 )
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: r in C /. 1
1 <= len C by A1, A2, A3, A4, Th65;
then A5: C . 1 = C /. 1 by FINSEQ_4:24;

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

suppose [.r,s.] in F ; :: thesis: r in C /. 1

then C = <*[.r,s.]*> by A1, A2, A3, A4, Def2;
then C /. 1 = [.r,s.] by FINSEQ_4:25;
hence r in C /. 1 by A4, XXREAL_1:1; :: thesis: verum

end;

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

then ex p being real number st
( r < p & p <= s & C . 1 = [.r,p.[ ) by A1, A2, A3, A4, Def2;
hence r in C /. 1 by A5, XXREAL_1:3; :: thesis: verum

end;

end;