let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA
for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) & ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) holds
(IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
let s be State of SCM+FSA; :: thesis: for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA
for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) & ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) holds
(IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
let J be Program of SCM+FSA; :: thesis: for Ig being good Program of SCM+FSA
for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) & ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) holds
(IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
let Ig be good Program of SCM+FSA; :: thesis: for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) & ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) holds
(IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
let f be FinSeq-Location ; :: thesis: ( ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) & ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) implies (IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f )
set I = Ig;
assume that
A1: ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) and
A2: ( J is parahalting or ( J is_halting_on IExec (Ig,p,s),p & J is_closed_on IExec (Ig,p,s),p ) ) ; :: thesis: (IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
A3: J is_halting_on IExec (Ig,p,s),p by A2, SCMFSA7B:19;
A4: Ig is_closed_on Initialized s,p by A1, SCMFSA7B:18;
A5: not f in dom (Start-At (((IC (IExec (J,p,(IExec (Ig,p,s))))) + (card Ig)),SCM+FSA)) by SCMFSA_2:103;
A6: J is_closed_on IExec (Ig,p,s),p by A2, SCMFSA7B:18;
Ig is_halting_on Initialized s,p by A1, SCMFSA7B:19;
then IExec ((Ig ';' J),p,s) = (IExec (J,p,(IExec (Ig,p,s)))) +* (Start-At (((IC (IExec (J,p,(IExec (Ig,p,s))))) + (card Ig)),SCM+FSA)) by A3, A4, A6, Th7;
hence (IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f by A5, FUNCT_4:11; :: thesis: verum