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 Th55:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
& r <= s & 1 <= n & n+1 <= len C implies ].lower_bound(C/.(n+1)),upper_bound(C
  /.n).[ is non empty
proof
  assume F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is
  connected & r <= s & 1 <= n & n+1 <= len C;
  then lower_bound(C/.(n+1)) < upper_bound(C/.n) by Def2;
  hence thesis by XXREAL_1:33;
end;
