let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st I does_not_destroy 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 I does_not_destroy 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: ( I does_not_destroy 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: I does_not_destroy 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: s . (IC (Comput (ProgramPart s),s,k)) does_not_destroy a by A1, SCMFSA7B:def 4;
Y: (ProgramPart (Comput (ProgramPart s),s,k)) /. (IC (Comput (ProgramPart s),s,k)) = (Comput (ProgramPart s),s,k) . (IC (Comput (ProgramPart s),s,k)) by AMI_1:150;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,k) by AMI_1:144;
(Comput (ProgramPart s),s,(k + 1)) . a = (Following (ProgramPart s),(Comput (ProgramPart s),s,k)) . a by AMI_1:14
.= (Exec (s . (IC (Comput (ProgramPart s),s,k))),(Comput (ProgramPart s),s,k)) . a by AMI_1:54, Y, T
.= s . a by A7, A9, SCMFSA7B:26 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A10: S₁[ 0 ] by AMI_1:13;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A10, A6); :: thesis: verum