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;
reserve n for Nat;

theorem Th12:
  for S be halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N
  for l being Nat
  for P being NAT-defined (the InstructionsF of S)-valued Function
   st l .--> halt S c= P
  for p being l-started PartState of S
   holds p is P-autonomic
proof
  let S be halting IC-Ins-separated
   non empty with_non-empty_values AMI-Struct over N;
  let l be Nat;
   set h = halt S;
   set p = Start-At(l,S);
  let P be NAT-defined (the InstructionsF of S)-valued Function such that
A1: l .--> halt S c= P;
  let p be l-started PartState of S;
   set I = p +* P;
  let Q1,Q2 be Instruction-Sequence of S such that
A2: P c= Q1 & P c= Q2;
  let s1,s2 be State of S;
    assume that
A3:  p c= s1 and
A4:  p c= s2;
    let i;
A5:  l .--> halt S c= Q1 & l .--> halt S c= Q2 by A1,A2,XBOOLE_1:1;
   hence Comput(Q1,s1,i)|dom  p
      = s1|dom  p by A3,Th11
     .=  p by A3,GRFUNC_1:23
     .= s2|dom  p by A4,GRFUNC_1:23
     .= Comput(Q2,s2,i)|dom  p by A4,Th11,A5;
end;
