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 & Initialized I c= s holds
for k being Element of NAT holds (Comput ((ProgramPart s),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 & Initialized I c= s holds
for k being Element of NAT holds (Comput ((ProgramPart s),s,k)) . a = s . a
let a be Int-Location ; :: thesis: ( not I destroys a & I is_closed_onInit s & Initialized I c= s implies for k being Element of NAT holds (Comput ((ProgramPart s),s,k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: ( not I is_closed_onInit s or not Initialized I c= s or for k being Element of NAT holds (Comput ((ProgramPart s),s,k)) . a = s . a )
defpred S₁[ Nat] means (Comput ((ProgramPart s),s,$1)) . a = s . a;
assume A2: I is_closed_onInit s ; :: thesis: ( not Initialized I c= s or for k being Element of NAT holds (Comput ((ProgramPart s),s,k)) . a = s . a )
assume A3: Initialized I c= s ; :: thesis: for k being Element of NAT holds (Comput ((ProgramPart s),s,k)) . a = s . a
then A4: s +* (Initialized I) = s by FUNCT_4:79;
A5: I c= s by A3, Th13;

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 ((ProgramPart s),s,k));
A8: IC (Comput ((ProgramPart s),s,k)) in dom I by A2, A4, Def4;
then s . (IC (Comput ((ProgramPart s),s,k))) = I . (IC (Comput ((ProgramPart s),s,k))) by A5, GRFUNC_1:8;
then s . (IC (Comput ((ProgramPart s),s,k))) in rng I by A8, FUNCT_1:def 5;
then A9: not s . (IC (Comput ((ProgramPart s),s,k))) destroys a by A1, SCMFSA7B:def 4;
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,k)) by AMI_1:123;
Y: (ProgramPart s) /. (IC (Comput ((ProgramPart s),s,k))) = (Comput ((ProgramPart s),s,k)) . (IC (Comput ((ProgramPart s),s,k))) by T, COMPOS_1:38;
(Comput ((ProgramPart s),s,(k + 1))) . a = (Following ((ProgramPart s),(Comput ((ProgramPart s),s,k)))) . a by EXTPRO_1:4
.= (Exec ((s . (IC (Comput ((ProgramPart s),s,k)))),(Comput ((ProgramPart s),s,k)))) . a by Y, AMI_1:54
.= s . a by A7, A9, SCMFSA7B:26 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A10: S₁[ 0 ] by EXTPRO_1:3;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A10, A6); :: thesis: verum