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;
reserve G for IntervalCoverPts of C;

theorem Th62:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s & 1 <= n & n+1 < len G implies G.(n+1) < upper_bound(C/.n)
proof
  assume F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is
  connected & r <= s & 1 <= n & n+1 < len G;
  then G.(n+1) in ].lower_bound(C/.(n+1)),upper_bound(C/.n).[ by Def3;
  hence thesis by XXREAL_1:4;
end;
