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 & len C = 1 holds
C = <*[.r,s.]*>
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 & len C = 1 holds
C = <*[.r,s.]*>
let C be IntervalCover of F; :: thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 implies C = <*[.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: C = <*[.r,s.]*>
A3: union (rng C) = [.r,s.] by A1, Def2;
not C is empty by A2;
then not rng C is empty ;
then 1 in dom C by FINSEQ_3:32;
then A4: C . 1 in rng C by FUNCT_1:def 3;
C . 1 = [.r,s.]

proof

thus for a being object st a in C . 1 holds
a in [.r,s.] by A3, A4, TARSKI:def 4; :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: [.r,s.] c= C . 1
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in [.r,s.] or a in C . 1 )
A5: dom C = {1} by A2, FINSEQ_1:2, FINSEQ_1:def 3;
assume a in [.r,s.] ; :: thesis: a in C . 1
then consider Z being set such that
A6: a in Z and
A7: Z in rng C by A3, TARSKI:def 4;
ex x being object st
( x in dom C & C . x = Z ) by A7, FUNCT_1:def 3;
hence a in C . 1 by A6, A5, TARSKI:def 1; :: thesis: verum

end;

hence C = <*[.r,s.]*> by A2, FINSEQ_1:40; :: thesis: verum