reserve x for set,
  D for non empty set,
  k, n for 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 Nat,
  s for State of S;
reserve ss for Element of product the_Values_of S;
reserve T for standard IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N;

theorem
 for N being with_zero set
 for S being IC-Ins-separated non empty with_non-empty_values AMI-Struct over N
 for F being finite preProgram of S
  holds F is really-closed iff
  for s being State of S st IC s in dom F
  for P being Instruction-Sequence of S st F c= P
   for k being Nat holds IC Comput(P,s,k) in dom F by Lm6;
