let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_onInit s,p & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT holds (Comput (p,s,k)) . a = s . a
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_onInit s,p & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT holds (Comput (p,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_onInit s,p & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT holds (Comput (p,s,k)) . a = s . a
let a be Int-Location ; :: thesis: ( not I destroys a & I is_closed_onInit s,p & Initialize ((intloc 0) .--> 1) c= s & I c= p implies for k being Element of NAT holds (Comput (p,s,k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: ( not I is_closed_onInit s,p or not Initialize ((intloc 0) .--> 1) c= s or not I c= p or for k being Element of NAT holds (Comput (p,s,k)) . a = s . a )
defpred S₁[ Nat] means (Comput (p,s,$1)) . a = s . a;
assume A2: I is_closed_onInit s,p ; :: thesis: ( not Initialize ((intloc 0) .--> 1) c= s or not I c= p or for k being Element of NAT holds (Comput (p,s,k)) . a = s . a )
assume Initialize ((intloc 0) .--> 1) c= s ; :: thesis: ( not I c= p or for k being Element of NAT holds (Comput (p,s,k)) . a = s . a )
then A3: Initialized s = s by FUNCT_4:98;
assume A4: I c= p ; :: thesis: for k being Element of NAT holds (Comput (p,s,k)) . a = s . a
then A5: p +* I = p by FUNCT_4:98;

A6: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A7: S₁[k] ; :: thesis: S₁[k + 1]
set l = IC (Comput (p,s,k));
A8: IC (Comput (p,s,k)) in dom I by A2, A3, Def4, A5;
then p . (IC (Comput (p,s,k))) = I . (IC (Comput (p,s,k))) by A4, GRFUNC_1:2;
then p . (IC (Comput (p,s,k))) in rng I by A8, FUNCT_1:def 3;
then A9: not p . (IC (Comput (p,s,k))) destroys a by A1, SCMFSA7B:def 4;
(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 A7, A9, SCMFSA7B:20 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A10: S₁[ 0 ] by EXTPRO_1:2;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A10, A6); :: thesis: verum