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 Th64:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s & 1 <= n & n < len C implies G.n <= lower_bound(C/.(n+1))
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r <= s;
  set w = n-'1;
  assume that
A3: 1 <= n and
A4: n < len C;
A5: n+1 <= len C by A4,NAT_1:13;
  per cases by A3,XXREAL_0:1;
  suppose
A6: n = 1;
    0+1 <= n+1 by XREAL_1:6;
    then
A7: C/.(n+1) is non empty by A1,A2,A5,Def2;
A8: G.1 = r by A1,A2,Def3;
A9: rng C c= F by A1,A2,Def2;
    1+1 <= len C by A4,A6,NAT_1:13;
    then
A10: 2 in dom C by FINSEQ_3:25;
    then C.2 in rng C by FUNCT_1:def 3;
    then C.2 in F by A9;
    then C/.2 in F by A10,PARTFUN1:def 6;
    then C/.(n+1) c= the carrier of Closed-Interval-TSpace(r,s) by A6;
    then
A11: C/.(n+1) c= [.r,s.] by A2,TOPMETR:18;
    then C/.(n+1) is bounded_below by XXREAL_2:44;
    then lower_bound(C/.(n+1)) in [.r,s.] by A7,A11,Th1;
    hence thesis by A6,A8,XXREAL_1:1;
  end;
  suppose
    1 < n;
    then
A12: 1-1 < n-1 by XREAL_1:9;
    then
A13: w = n-1 by XREAL_0:def 2;
    then
A14: 0+1 <= w by A12,NAT_1:13;
    len G = len C + 1 by A1,A2,Def3;
    then
A15: n+1 < len G - 1+1 by A4,XREAL_1:6;
    n-1 < n-0 by XREAL_1:15;
    then w+1 < n+1 by A13,XREAL_1:6;
    then w+1 < len G by A15,XXREAL_0:2;
    then
A16: G.(w+1) < upper_bound(C/.w) by A1,A2,A14,Th62;
    n+1 <= len C by A4,NAT_1:13;
    then upper_bound(C/.w) <= lower_bound(C/.(w+2)) by A1,A2,A13,A14,Def2;
    hence thesis by A13,A16,XXREAL_0:2;
  end;
end;
