let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being InitHalting keepInt0_1 Program of SCM+FSA
for a being read-write Int-Location st not I destroys a holds
(Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1
let s be State of SCM+FSA; :: thesis: for I being InitHalting keepInt0_1 Program of SCM+FSA
for a being read-write Int-Location st not I destroys a holds
(Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1
let I be InitHalting keepInt0_1 Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a holds
(Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1
let a be read-write Int-Location ; :: thesis: ( not I destroys a implies (Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1 )
assume A1: not I destroys a ; :: thesis: (Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1
set s0 = Initialized s;
set p0 = p;
set s1 = Initialize (Initialized s);
set p1 = p +* (I ';' (SubFrom (a,(intloc 0))));
AA: a <> IC by SCMFSA_2:56;
dom (Start-At (0,SCM+FSA)) = {(IC )} by FUNCOP_1:13;
then A2: not a in dom (Start-At (0,SCM+FSA)) by AA, TARSKI:def 1;
(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)) . a = (Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,s)))) . a by Th33
.= ((IExec (I,p,s)) . a) - ((IExec (I,p,s)) . (intloc 0)) by SCMFSA_2:65
.= ((IExec (I,p,s)) . a) - 1 by Th17
.= ((Comput ((p +* I),(Initialize (Initialized s)),0)) . a) - 1 by A1, Th62
.= ((Initialize (Initialized s)) . a) - 1 by EXTPRO_1:2
.= ((Initialized s) . a) - 1 by A2, FUNCT_4:11 ;
hence (Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = ((Initialized s) . a) - 1 by Th61
.= (s . a) - 1 by SCMFSA6C:3 ;
:: thesis: verum