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
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected & r <=
s & n in dom C & m in dom C & n < m implies upper_bound(C/.n) <= upper_bound(C
  /.m)
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is
  connected & r <= s and
A2: n in dom C and
A3: m in dom C & n < m;
  defpred P[Nat] means $1 in dom C & n < $1 implies upper_bound(C/.
  n) <= upper_bound(C/.$1);
A4: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A5: P[k] and
A6: k+1 in dom C and
A7: n < k+1;
    per cases by A6,TOPREALA:2;
    suppose
      k = 0;
      then n = 0 by A7,NAT_1:13;
      hence thesis by A2,FINSEQ_3:24;
    end;
    suppose
A8:   k in dom C;
A9:   k+1 <= len C by A6,FINSEQ_3:25;
A10:  n <= k by A7,NAT_1:13;
      1 <= k by A8,FINSEQ_3:25;
      then upper_bound(C/.k) <= upper_bound(C/.(k+1)) by A1,A9,Def2;
      hence thesis by A5,A8,A10,XXREAL_0:1,2;
    end;
  end;
A11: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A11,A4);
  hence thesis by A3;
end;
