reserve a, b, r, s for Real;
reserve n, m for Nat,
  F for Subset-Family of Closed-Interval-TSpace (r,s);
reserve C for IntervalCover of F;

theorem
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected & r <=
  s implies upper_bound(C/.len C) = s
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r <= s;
  1 <= len C by A1,A2,Th51;
  then
A3: C.len C = C/.len C by FINSEQ_4:15;
  per cases;
  suppose
    [.r,s.] in F;
    then C = <*[.r,s.]*> by A1,A2,Def2;
    then C/.1 = [.r,s.] & len C = 1 by FINSEQ_1:39,FINSEQ_4:16;
    hence thesis by A2,JORDAN5A:19;
  end;
  suppose
    not [.r,s.] in F;
    then
    ex p being Real st r <= p & p < s & C.len C = ].p,s.] by A1,A2,Def2;
    hence thesis by A3,Th7;
  end;
end;
