let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
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),(s +* (Initialize I)),k)) . a = s . a
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: 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),(s +* (Initialize I)),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),(s +* (Initialize I)),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),(s +* (Initialize I)),k)) . a = s . a )
A1: ProgramPart I = I by RELAT_1:209;
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),(s +* (Initialize I)),k)) . a = s . a )
defpred S₁[ Nat] means (Comput ((P +* I),(s +* (Initialize I)),$1)) . a = s . a;
A3: I c= P +* I by FUNCT_4:26;
assume A4: I is_closed_on s,P ; :: thesis: for k being Element of NAT holds (Comput ((P +* I),(s +* (Initialize I)),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),(s +* (Initialize I)),k));
A7: IC (Comput ((P +* I),(s +* (Initialize I)),k)) in dom I by A4, Def7, A1;
then (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),k))) = I . (IC (Comput ((P +* I),(s +* (Initialize I)),k))) by GRFUNC_1:8, A3;
then (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),k))) in rng I by A7, FUNCT_1:def 5;
then A8: not (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),k))) destroys a by A2, Def4;
A9: dom (P +* I) = NAT by PARTFUN1:def 4;
(Comput ((P +* I),(s +* (Initialize I)),(k + 1))) . a = (Following ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),k)))) . a by EXTPRO_1:4
.= (Exec (((P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),k)))),(Comput ((P +* I),(s +* (Initialize I)),k)))) . a by A9, PARTFUN1:def 8
.= (Comput ((P +* I),(s +* (Initialize I)),k)) . a by A8, Th26
.= s . a by A6 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A10: not a in dom (Initialize I) by SCMFSA6B:12;
(Comput ((P +* I),(s +* (Initialize I)),0)) . a = (s +* (Initialize I)) . a by EXTPRO_1:3
.= s . a by A10, FUNCT_4:12 ;
then A11: S₁[ 0 ] ;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A11, A5); :: thesis: verum