let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being keeping_0 Program of SCM+FSA st I is_halting_on Initialized s,P holds
( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) )
set a = intloc 0;
let s be State of SCM+FSA; :: thesis: for I being keeping_0 Program of SCM+FSA st I is_halting_on Initialized s,P holds
( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) )
set D = Data-Locations ;
set A = NAT ;
let I be keeping_0 Program of SCM+FSA; :: thesis: ( I is_halting_on Initialized s,P implies ( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) ) )
set s0 = Initialized s;
set s1 = Initialize (Initialized s);
set P1 = P +* I;
A1: I c= P +* I by FUNCT_4:25;
A4: DataPart (Initialized s) = DataPart (Initialize (Initialized s)) by MEMSTR_0:79;

A5: now

let k be Element of NAT ; :: thesis: (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1
thus (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = (Initialize (Initialized s)) . (intloc 0) by A1, SCMFSA6B:def 4
.= (Initialized s) . (intloc 0) by A4, SCMFSA6A:7
.= 1 by SCMFSA6A:38 ; :: thesis: verum

end;

assume I is_halting_on Initialized s,P ; :: thesis: ( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) )
then A7: P +* I halts_on Initialize (Initialized s) by SCMFSA7B:def 7;
thus (IExec (I,P,s)) . (intloc 0) = (Result ((P +* I),(Initialize (Initialized s)))) . (intloc 0) by MEMSTR_0:44
.= (Result ((P +* I),(Initialize (Initialized s)))) . (intloc 0)
.= (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . (intloc 0) by A7, EXTPRO_1:23
.= 1 by A5 ; :: thesis: for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1
let k be Element of NAT ; :: thesis: (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1
thus (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 by A5; :: thesis: verum