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),(Initialize s),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),(Initialize s),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),(Initialize s),k)) . a = s . a
let a be Int-Location; :: thesis: ( not I destroys a implies for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
defpred S₁[ Nat] means (Comput ((P +* I),(Initialize s),$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),(Initialize s),k));
IC (Initialize s) = 0 by MEMSTR_0:47;
then IC (Initialize s) in dom I by AFINSQ_1:65;
then A5: IC (Comput ((P +* I),(Initialize s),k)) in dom I by A2, AMISTD_1:21;
then (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) = I . (IC (Comput ((P +* I),(Initialize s),k))) by A2, GRFUNC_1:2;
then (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) in rng I by A5, FUNCT_1:def 3;
then A6: not (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) destroys a by A1;
A7: dom (P +* I) = NAT by PARTFUN1:def 2;
(Comput ((P +* I),(Initialize s),(k + 1))) . a = (Following ((P +* I),(Comput ((P +* I),(Initialize s),k)))) . a by EXTPRO_1:3
.= (Exec (((P +* I) . (IC (Comput ((P +* I),(Initialize s),k)))),(Comput ((P +* I),(Initialize s),k)))) . a by A7, PARTFUN1:def 6
.= (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) . a by A6, Th19
.= s . a by A4 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A8: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
(Comput ((P +* I),(Initialize s),0)) . a = (Initialize s) . a
.= s . a by A8, FUNCT_4:11 ;
then A9: S₁[ 0 ] ;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A9, A3); :: thesis: verum