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 Th50:
  for F being Subset-Family of Closed-Interval-TSpace(r,r), C
being IntervalCover of F holds F is Cover of Closed-Interval-TSpace(r,r) & F is
  open connected implies C = <*[.r,r.]*>
proof
  set L = Closed-Interval-TSpace(r,r);
  let F be Subset-Family of L, C be IntervalCover of F;
  assume that
A1: F is Cover of L and
A2: F is open & F is connected;
A3: [.r,r.] = {r} by XXREAL_1:17;
  the carrier of L = [.r,r.] by TOPMETR:18;
  then r in the carrier of L by A3,TARSKI:def 1;
  then {r} in F by A1,Th46;
  hence thesis by A1,A2,A3,Def2;
end;
