let p be Instruction-Sequence of SCM+FSA; for s being State of SCM+FSA
for f being FinSeq-Location
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_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; for f being FinSeq-Location
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_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 ; for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_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 really-closed Program of SCM+FSA; for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_on IExec (Ig,p,s),p ) holds
(IExec ((Ig ";" J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
let J be really-closed Program of SCM+FSA; ( ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_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 )
and
A2:
( J is parahalting or J is_halting_on IExec (Ig,p,s),p )
; (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:
not f in dom (Start-At (((IC (IExec (J,p,(IExec (Ig,p,s))))) + (card Ig)),SCM+FSA))
by SCMFSA_2:103;
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, Th5;
hence
(IExec ((Ig ";" J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f
by A4, FUNCT_4:11; verum