reserve x for set,
  D for non empty set,
  k, n for Element of NAT,
  z for Nat;
reserve N for with_zero set,
  S for
    IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  i for Element of the InstructionsF of S,
  l, l1, l2, l3 for Element of NAT,
  s for State of S;
reserve ss for Element of product the_Values_of S;
reserve T for weakly_standard
 IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N;

theorem
  for N being with_zero set,
      S being IC-Ins-separated
       non empty with_non-empty_values AMI-Struct over N,
  l1, l2 being Element of NAT st SUCC(l1,S) = NAT
   holds l1 <= l2, S
proof
  let N be with_zero set,
  S be IC-Ins-separated non
empty with_non-empty_values AMI-Struct over N, l1, l2 be Element of NAT
  such that
A1: SUCC(l1,S) = NAT;
  defpred P[set,set] means ($1 = 1 implies $2 = l1) & ($1 = 2 implies $2 = l2);
A2: for n being Nat st n in Seg 2 ex d being Element of NAT st P[n,d]
  proof
    let n be Nat;
    assume
A3: n in Seg 2;
    per cases by A3,FINSEQ_1:2,TARSKI:def 2;
    suppose
A4:   n = 1;
      reconsider l1 as Element of NAT;
      take l1;
      thus thesis by A4;
    end;
    suppose
A5:   n = 2;
      reconsider l2 as Element of NAT;
      take l2;
      thus thesis by A5;
    end;
  end;
  consider f being FinSequence of NAT such that
A6: len f = 2 and
A7: for n being Nat st n in Seg 2 holds P[n,f/.n] from FINSEQ_4:sch 1(A2);
A8: 1 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then
A9: f/.1 = l1 by A7;
  reconsider f as non empty FinSequence of NAT by A6;
  take f;
  2 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  hence f/.1 = l1 & f/.len f = l2 by A6,A7,A8;
  let n be Element of NAT;
  assume
A10: 1 <= n;
  assume n < len f;
  then n < 1+1 by A6;
  then n <= 1 by NAT_1:13;
  then n = 1 by A10,XXREAL_0:1;
  hence thesis by A1,A9;
end;
