let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_on s,P holds
for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_on s,P holds
for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let I be Program of SCM+FSA; :: thesis: for a being Int-Location st not I destroys a & I is_closed_on s,P holds
for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let a be Int-Location ; :: thesis: ( not I destroys a & I is_closed_on s,P implies for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a )
assume A2: not I destroys a ; :: thesis: ( not I is_closed_on s,P or for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a )
defpred S₁[ Nat] means (Comput ((P +* I),(Initialize s),$1)) . a = s . a;
A3: I c= P +* I by FUNCT_4:25;
assume A4: I is_closed_on s,P ; :: thesis: for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . a = s . a

A5: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]
set l = IC (Comput ((P +* I),(Initialize s),k));
A7: IC (Comput ((P +* I),(Initialize s),k)) in dom I by A4, Def7;
then (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) = I . (IC (Comput ((P +* I),(Initialize s),k))) by A3, GRFUNC_1:2;
then (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) in rng I by A7, FUNCT_1:def 3;
then A8: not (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) destroys a by A2, Def4;
A9: 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 A9, PARTFUN1:def 6
.= (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) . a by A8, Th26
.= s . a by A6 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A10: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
(Comput ((P +* I),(Initialize s),0)) . a = (Initialize s) . a by EXTPRO_1:2
.= s . a by A10, FUNCT_4:11 ;
then A11: S₁[ 0 ] ;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A11, A5); :: thesis: verum