let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being really-closed Program of SCM+FSA
for a being Int-Location st not I destroys a & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Nat holds (Comput (p,s,k)) . a = s . a
let s be State of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA
for a being Int-Location st not I destroys a & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Nat holds (Comput (p,s,k)) . a = s . a
let I be really-closed Program of SCM+FSA; :: thesis: for a being Int-Location st not I destroys a & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Nat holds (Comput (p,s,k)) . a = s . a
let a be Int-Location; :: thesis: ( not I destroys a & Initialize ((intloc 0) .--> 1) c= s & I c= p implies for k being Nat holds (Comput (p,s,k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: ( not Initialize ((intloc 0) .--> 1) c= s or not I c= p or for k being Nat holds (Comput (p,s,k)) . a = s . a )
defpred S₁[ Nat] means (Comput (p,s,$1)) . a = s . a;
assume Initialize ((intloc 0) .--> 1) c= s ; :: thesis: ( not I c= p or for k being Nat holds (Comput (p,s,k)) . a = s . a )
then A2: Initialized s = s by FUNCT_4:98;
assume A3: I c= p ; :: thesis: for k being Nat holds (Comput (p,s,k)) . a = s . a

A4: 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 A5: S₁[k] ; :: thesis: S₁[k + 1]
set l = IC (Comput (p,s,k));
IC s = 0 by A2, MEMSTR_0:def 11;
then IC s in dom I by AFINSQ_1:65;
then A6: IC (Comput (p,s,k)) in dom I by AMISTD_1:21, A3;
then p . (IC (Comput (p,s,k))) = I . (IC (Comput (p,s,k))) by A3, GRFUNC_1:2;
then p . (IC (Comput (p,s,k))) in rng I by A6, FUNCT_1:def 3;
then A7: not p . (IC (Comput (p,s,k))) destroys a by A1;
(Comput (p,s,(k + 1))) . a = (Following (p,(Comput (p,s,k)))) . a by EXTPRO_1:3
.= (Exec ((p . (IC (Comput (p,s,k)))),(Comput (p,s,k)))) . a by PBOOLE:143
.= s . a by A5, A7, SCMFSA7B:20 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A8: S₁[ 0 ] ;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A8, A4); :: thesis: verum