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;
reserve N for with_zero non empty set;

theorem
  i <= j implies for S being halting IC-Ins-separated
non empty with_non-empty_values AMI-Struct over N
 for P being (the InstructionsF of S)-valued NAT-defined Function
 for s being State of S st P halts_at IC Comput(P,s,i)
   holds Comput(P,s,j) = Comput(P,s,i)
proof
  assume
A1: i <= j;
  let S be halting IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N,
  P be (the InstructionsF of S)-valued NAT-defined Function,
  s be State of S;
  assume
A2: P halts_at IC Comput(P,s,i);
  then P halts_at IC Comput(P,s,j) by A1,Th19;
  hence Comput(P,s,j) = Result(P,s) by Th18
    .= Comput(P,s,i) by A2,Th18;
end;
