reserve N for with_zero set;
reserve N for with_zero set;
reserve x,y,z,A,B for set,
  f,g,h for Function,
  i,j,k for Nat;
reserve S for IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve N for non empty with_zero set,
 S for IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  s for State of S;

theorem Th5:
  i <= j implies
 for N for S being halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N
  for p being NAT-defined (the InstructionsF of S)-valued Function
  for s being State of S st CurInstr(p,Comput(p,s,i)) = halt S
 holds Comput(p,s,j) = Comput(p,s,i)
proof
  assume i <= j;
  then consider k being Nat such that
A1: j = i + k by NAT_1:10;
  reconsider k as Nat;
A2: j = i + k by A1;
  let N;
  let S be halting IC-Ins-separated non empty
       with_non-empty_values AMI-Struct over N;
  let p be NAT-defined (the InstructionsF of S)-valued Function;
  let s be State of S such that
A3: CurInstr(p,Comput(p,s,i))
 = halt S;
  defpred P[Nat] means Comput(p,s,i+$1)
   = Comput(p,s,i);
A4: now
    let k;
    assume
A5: P[k];
    Comput(p,s,i+(k+1)) = Comput(p,s,i+k+1)
      .= Following(p,Comput(p,s,i+k)) by Th3
      .= Comput(p,s,i) by A3,A5,Def3;
    hence P[k+1];
  end;
A6: P[ 0];
  for k holds P[k] from NAT_1:sch 2(A6,A4);
  hence thesis by A2;
end;
