let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for f being FinSeq-Location
for p being FinSequence of INT holds
( (IExec ((f := p),P,s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),P,s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),P,s)) . g = s . g ) )
let s be State of SCM+FSA; :: thesis: for f being FinSeq-Location
for p being FinSequence of INT holds
( (IExec ((f := p),P,s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),P,s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),P,s)) . g = s . g ) )
let f be FinSeq-Location ; :: thesis: for p being FinSequence of INT holds
( (IExec ((f := p),P,s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),P,s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),P,s)) . g = s . g ) )
let p be FinSequence of INT ; :: thesis: ( (IExec ((f := p),P,s)) . f = p & ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),P,s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),P,s)) . g = s . g ) )
A1: (s +* (Initialize ((intloc 0) .--> 1))) . (intloc 0) = (Initialize ((intloc 0) .--> 1)) . (intloc 0) by FUNCT_4:13, SCMFSA6A:41
.= 1 by SCMFSA6A:43 ;
reconsider s1 = s +* (Initialize ((intloc 0) .--> 1)) as 0 -started State of SCM+FSA ;
A2: f := p c= P +* (f := p) by FUNCT_4:25;
thus (IExec ((f := p),P,s)) . f = (Result ((P +* (f := p)),s1)) . f
.= p by A1, A2, SCMFSA_7:7 ; :: thesis: ( ( for a being read-write Int-Location st a <> intloc 1 & a <> intloc 2 holds
(IExec ((f := p),P,s)) . a = s . a ) & ( for g being FinSeq-Location st g <> f holds
(IExec ((f := p),P,s)) . g = s . g ) )

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

let a be read-write Int-Location ; :: thesis: ( a <> intloc 1 & a <> intloc 2 implies (IExec ((f := p),P,s)) . a = s . a )
( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then B6: not a in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA6A:42, TARSKI:def 2;
assume A3: ( a <> intloc 1 & a <> intloc 2 ) ; :: thesis: (IExec ((f := p),P,s)) . a = s . a
thus (IExec ((f := p),P,s)) . a = (Result ((P +* (f := p)),(s +* (Initialize ((intloc 0) .--> 1))))) . a
.= s1 . a by A1, A2, A3, SCMFSA_7:7
.= s . a by B6, FUNCT_4:11 ; :: thesis: verum

end;

let g be FinSeq-Location ; :: thesis: ( g <> f implies (IExec ((f := p),P,s)) . g = s . g )
assume A5: g <> f ; :: thesis: (IExec ((f := p),P,s)) . g = s . g
( g <> intloc 0 & g <> IC ) by SCMFSA_2:57, SCMFSA_2:58;
then B6: not g in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA6A:42, TARSKI:def 2;
thus (IExec ((f := p),P,s)) . g = (Result ((P +* (f := p)),(s +* (Initialize ((intloc 0) .--> 1))))) . g
.= s1 . g by A1, A2, A5, SCMFSA_7:7
.= s . g by B6, FUNCT_4:11 ; :: thesis: verum