reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th5:
  for h being real-valued FinSequence st h is increasing holds for
  i,j being Nat st i<j & 1<=i & j<=len h holds h.i < h.j
proof
  let h be real-valued FinSequence;
  assume
A1: h is increasing;
  let i,j be Nat;
  assume that
A2: i<j and
A3: 1<=i and
A4: j<=len h;
  defpred P[Nat] means i+1+$1<=len h implies h.i<h.(i+1+$1);
A5: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A6: P[k];
    i+1+(k+1)<=len h implies h.i<h.(i+1+(k+1))
    proof
A7:   i+1<=i+1+k by NAT_1:11;
      i<i+1 by XREAL_1:29;
      then i<i+1+k by A7,XXREAL_0:2;
      then
A8:   1<i+1+k by A3,XXREAL_0:2;
      k<k+1 by XREAL_1:29;
      then
A9:   i+1+k<i+1+(k+1) by XREAL_1:6;
      assume
A10:  i+1+(k+1)<=len h;
      then i+1+k<len h by A9,XXREAL_0:2;
      then h.(i+1+k)<h.(i+1+k+1) by A1,A8;
      hence h.i<h.(i+1+(k+1)) by A6,A10,A9,XXREAL_0:2;
    end;
    hence P[k+1];
  end;
  i<len h by A2,A4,XXREAL_0:2;
  then
A11: P[0] by A1,A3;
  for k being Nat holds P[k] from NAT_1:sch 2(A11,A5);
  then
A12: i+1+(j-'(i+1))<=len h implies h.i<h.(i+1+(j-'(i+1)));
  i+1<=j by A2,NAT_1:13;
  then j-'(i+1)=j-(i+1) by XREAL_1:233;
  hence h.i<h.j by A4,A12;
end;
