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 Th63:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s & 1 < n & n <= len C implies lower_bound(C/.n) < G.n
proof
  set w = n-'1;
  assume
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is
  connected & r <= s;
  then
A2: len G = len C + 1 by Def3;
  assume that
A3: 1 < n and
A4: n <= len C;
A5: n < len C + 1 by A4,NAT_1:13;
  1-1 <= n-1 by A3,XREAL_1:9;
  then
A6: w = n-1 by XREAL_0:def 2;
  then n = w+1;
  then 1 <= w by A3,NAT_1:13;
  then G.(w+1) in ].lower_bound(C/.(w+1)),upper_bound(C/.w).[ by A1,A2,A6,A5
,Def3;
  hence thesis by A6,XXREAL_1:4;
end;
