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 Th65:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r < s implies G is increasing
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r < s;
  let m, n be Nat such that
A3: m in dom G & n in dom G & m < n;
  defpred P[Nat] means m < $1 & m in dom G & $1 in dom G implies G.m < G.$1;
A4: for k being Nat st P[k] holds P[k+1]
  proof
A5: len G = len C + 1 by A1,A2,Def3;
    let k be Nat such that
A6: P[k] and
A7: m < k+1 and
A8: m in dom G and
A9: k+1 in dom G;
A10: 1 <= m by A8,FINSEQ_3:25;
A11: k+1 <= len G by A9,FINSEQ_3:25;
    k+0 <= k+1 by XREAL_1:6;
    then
A12: k <= len G by A11,XXREAL_0:2;
A13: m <= k by A7,NAT_1:13;
    then
A14: 1 <= k by A10,XXREAL_0:2;
    per cases by A14,A11,XXREAL_0:1;
    suppose that
A15:  1 < k and
A16:  k+1 < len G;
      G.(k+1) in ].lower_bound(C/.(k+1)),upper_bound(C/.k).[ by A1,A2,A15,A16
,Def3;
      then
A17:  lower_bound(C/.(k+1)) < G.(k+1) by XXREAL_1:4;
      k < len C by A5,A16,XREAL_1:6;
      then G.k <= lower_bound(C/.(k+1)) by A1,A2,A15,Th64;
      then G.k < G.(k+1) by A17,XXREAL_0:2;
      hence thesis by A6,A8,A13,A12,A15,FINSEQ_3:25,XXREAL_0:1,2;
    end;
    suppose
A18:  k = 1;
A19:  1 <= len C by A1,A2,Th51;
A20:  m <= 1 by A7,A18,NAT_1:13;
      per cases by A19,XXREAL_0:1;
      suppose
A21:    1 < len C;
        then 1+1 <= len C by NAT_1:13;
        then
A22:    lower_bound(C/.2) < G.2 by A1,A2,Th63;
        G.1 <= lower_bound(C/.(1+1)) by A1,A2,A21,Th64;
        then G.1 < G.2 by A22,XXREAL_0:2;
        hence thesis by A10,A18,A20,XXREAL_0:1;
      end;
      suppose
        1 = len C;
        then G = <*r,s*> by A1,A2,Th61;
        then G.1 = r & G.2 = s;
        hence thesis by A2,A10,A18,A20,XXREAL_0:1;
      end;
    end;
    suppose
A23:  k+1 = len G;
      then
A24:  G.(k+1) = s by A1,A2,Def3;
      per cases by A10,XXREAL_0:1;
      suppose
A25:    1 < m;
        set z = m-'1;
        1-1 <= m-1 by A10,XREAL_1:9;
        then
A26:    z = m-1 by XREAL_0:def 2;
        then
A27:    z+1 < len G by A7,A23;
        then
A28:    z <= len C by A5,XREAL_1:6;
        1+1 <= m by A25,NAT_1:13;
        then
A29:    1+1-1 <= m-1 by XREAL_1:9;
        then
A30:    1 <= z by XREAL_0:def 2;
        then
A31:    C/.z is non empty by A1,A2,A28,Def2;
A32:    rng C c= F by A1,A2,Def2;
A33:    z in dom C by A30,A28,FINSEQ_3:25;
        then C.z in rng C by FUNCT_1:def 3;
        then C.z in F by A32;
        then C/.z in F by A33,PARTFUN1:def 6;
        then C/.z c= the carrier of Closed-Interval-TSpace(r,s);
        then
A34:    C/.z c= [.r,s.] by A2,TOPMETR:18;
        then C/.z is bounded_above by XXREAL_2:43;
        then upper_bound(C/.z) in [.r,s.] by A34,A31,Th2;
        then
A35:    upper_bound(C/.z) <= s by XXREAL_1:1;
        G.m < upper_bound(C/.z) by A1,A2,A26,A29,A27,Th62;
        hence thesis by A24,A35,XXREAL_0:2;
      end;
      suppose
        m = 1;
        hence thesis by A1,A2,A24,Def3;
      end;
    end;
  end;
A36: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A36,A4);
  hence thesis by A3;
end;
