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 Th14:
 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 k being Nat holds IC Comput(F,s,k) in dom F
 proof let N be with_zero set;
  let S be IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N;
  let F be finite preProgram of S;
  thus F is really-closed implies
   for s being State of S st IC s in dom F
    for k being Nat holds IC Comput(F,s,k) in dom F
   proof assume
A1: F is really-closed;
     let s be State of S such that
A2: IC s in dom F;
     defpred P[Nat] means IC Comput(F,s,$1) in dom F;
A3: now
       let k be Nat such that
A4: P[k];
       reconsider t = Comput(F,s,k)
        as Element of product the_Values_of S by CARD_3:107;
       set l = IC Comput(F,s,k);
A5: IC Following(F,t) in NIC(F/.l,l);
A6: Comput(F,s,k+1) = Following(F,t) by EXTPRO_1:3;
       NIC(F/.l, l) c= dom F by A1,A4;
       hence P[k+1] by A5,A6;
     end;
A7: P[0] by A2;
     thus for k being Nat holds P[k] from NAT_1:sch 2(A7,A3);
   end;
  assume
A8: for s being State of S st IC s in dom F
     for k being Nat holds IC Comput(F,s,k) in dom F;
  let l being Nat such that
A9: l in dom F;
  let x be object;
  assume x in NIC(F/.l, l);
   then consider ss being Element of product the_Values_of S
    such that
A10: x = IC Exec(F/.l,ss) and
A11: IC ss = l;
   reconsider ss as State of S;
   IC Comput(F,ss,0+1) = IC Following(F,Comput(F,ss,0)) by EXTPRO_1:3
      .= IC Following(F,ss)
      .= IC Exec(F/.IC ss,ss);
  hence x in dom F by A10,A11,A8,A9;
 end;
