let s be State of SCM+FSA; :: thesis: for f being FinSeq-Location
for p being FinSequence of INT holds
( (IExec ((f := p),s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),s)) . g = s . g ) )
let f be FinSeq-Location ; :: thesis: for p being FinSequence of INT holds
( (IExec ((f := p),s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),s)) . g = s . g ) )
let p be FinSequence of INT ; :: thesis: ( (IExec ((f := p),s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),s)) . g = s . g ) )
A1: IExec ((f := p),s) = (Result ((ProgramPart (s +* (Initialized (f := p)))),(s +* (Initialized (f := p))))) +* (s | NAT) by SCMFSA6B:def 1;
intloc 0 in dom (Initialized (f := p)) by SCMFSA6A:45;
then A2: (s +* (Initialized (f := p))) . (intloc 0) = (Initialized (f := p)) . (intloc 0) by FUNCT_4:14
.= 1 by SCMFSA6A:46 ;
IC SCM+FSA in dom (Initialized (f := p)) by SCMFSA6A:24;
then (s +* (Initialized (f := p))) . (IC SCM+FSA) = (Initialized (f := p)) . (IC SCM+FSA) by FUNCT_4:14
.= 0 by SCMFSA6A:46 ;
then IC (s +* (Initialized (f := p))) = 0 ;
then reconsider s1 = s +* (Initialized (f := p)) as 0 -started State of SCM+FSA by COMPOS_1:def 20;
( Initialized (f := p) c= s +* (Initialized (f := p)) & f := p c= Initialized (f := p) ) by FUNCT_4:26, SCMFSA6A:26;
then A4: f := p c= s +* (Initialized (f := p)) by XBOOLE_1:1;
not f in NAT by SCMFSA_2:85;
then not f in (dom s) /\ NAT by XBOOLE_0:def 4;
then not f in dom (s | NAT) by RELAT_1:90;
hence (IExec ((f := p),s)) . f = (Result ((ProgramPart s1),s1)) . f by A1, FUNCT_4:12
.= p by A2, A4, SCMFSA_7:39 ;
:: thesis: ( ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),s)) . g = s . g ) )

hereby :: thesis: for g being FinSeq-Location st g <> f holds
(IExec ((f := p),s)) . g = s . g

let a be read-write Int-Location ; :: thesis: ( a <> intloc 1 & a <> intloc 2 implies (IExec ((f := p),s)) . a = s . a )
assume A6: ( a <> intloc 1 & a <> intloc 2 ) ; :: thesis: (IExec ((f := p),s)) . a = s . a
A7: not a in dom (Initialized (f := p)) by SCMFSA6A:48;
not a in NAT by SCMFSA_2:84;
then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then not a in dom (s | NAT) by RELAT_1:90;
hence (IExec ((f := p),s)) . a = (Result ((ProgramPart (s +* (Initialized (f := p)))),(s +* (Initialized (f := p))))) . a by A1, FUNCT_4:12
.= s1 . a by A2, A4, A6, SCMFSA_7:39
.= s . a by A7, FUNCT_4:12 ;
:: thesis: verum

end;

let g be FinSeq-Location ; :: thesis: ( g <> f implies (IExec ((f := p),s)) . g = s . g )
assume A9: g <> f ; :: thesis: (IExec ((f := p),s)) . g = s . g
A10: not g in dom (Initialized (f := p)) by SCMFSA6A:49;
not g in NAT by SCMFSA_2:85;
then not g in (dom s) /\ NAT by XBOOLE_0:def 4;
then not g in dom (s | NAT) by RELAT_1:90;
hence (IExec ((f := p),s)) . g = (Result ((ProgramPart (s +* (Initialized (f := p)))),(s +* (Initialized (f := p))))) . g by A1, FUNCT_4:12
.= s1 . g by A2, A4, A9, SCMFSA_7:39
.= s . g by A10, FUNCT_4:12 ;
:: thesis: verum