let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let s be State of SCM+FSA; :: thesis: for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let I be really-closed MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let a be read-write Int-Location; :: thesis: ( I is_halting_onInit s,P & s . a > 0 implies for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )
set s0 = Initialized s;
set IA = Start-At (0,SCM+FSA);
assume I is_halting_onInit s,P ; :: thesis: ( not s . a > 0 or for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )
then A1: P +* I halts_on Initialized s ;
Initialized s = Initialize (Initialized s) by MEMSTR_0:44;
then A2: I is_halting_on Initialized s,P by A1, SCMFSA7B:def 7;
assume s . a > 0 ; :: thesis: for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
then A3: (Initialized s) . a > 0 by SCMFSA_M:37;

hereby :: thesis: verum

let k be Nat; :: thesis: ( k <= (LifeSpan ((P +* I),(Initialized s))) + 2 implies IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )
A4: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;
assume k <= (LifeSpan ((P +* I),(Initialized s))) + 2 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
hence IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) by A4, A2, A3, SCMFSA_9:42; :: thesis: verum

end;