let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being really-closed Program of
for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(),k)) . a = s . a

let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed Program of
for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(),k)) . a = s . a

let I be really-closed Program of ; :: thesis: for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(),k)) . a = s . a

let a be Int-Location; :: thesis: ( not I destroys a implies for k being Nat holds (Comput ((P +* I),(),k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: for k being Nat holds (Comput ((P +* I),(),k)) . a = s . a
defpred S1[ Nat] means (Comput ((P +* I),(),\$1)) . a = s . a;
A2: I c= P +* I by FUNCT_4:25;
A3: now :: thesis: for k being Nat st S1[k] holds
S1[k + 1]
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A4: S1[k] ; :: thesis: S1[k + 1]
set l = IC (Comput ((P +* I),(),k));
IC () = 0 by MEMSTR_0:47;
then IC () in dom I by AFINSQ_1:65;
then A5: IC (Comput ((P +* I),(),k)) in dom I by ;
then (P +* I) . (IC (Comput ((P +* I),(),k))) = I . (IC (Comput ((P +* I),(),k))) by ;
then (P +* I) . (IC (Comput ((P +* I),(),k))) in rng I by ;
then A6: not (P +* I) . (IC (Comput ((P +* I),(),k))) destroys a by A1;
A7: dom (P +* I) = NAT by PARTFUN1:def 2;
(Comput ((P +* I),(),(k + 1))) . a = (Following ((P +* I),(Comput ((P +* I),(),k)))) . a by EXTPRO_1:3
.= (Exec (((P +* I) . (IC (Comput ((P +* I),(),k)))),(Comput ((P +* I),(),k)))) . a by
.= (Comput ((P +* I),(s +* ()),k)) . a by
.= s . a by A4 ;
hence S1[k + 1] ; :: thesis: verum
end;
A8: not a in dom () by SCMFSA_2:102;
(Comput ((P +* I),(),0)) . a = () . a
.= s . a by ;
then A9: S1[ 0 ] ;
thus for k being Nat holds S1[k] from NAT_1:sch 2(A9, A3); :: thesis: verum