let p be Instruction-Sequence of SCM+FSA; for s being State of SCM+FSA
for j being sequential Instruction of SCM+FSA
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
let s be State of SCM+FSA; for j being sequential Instruction of SCM+FSA
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
let j be sequential Instruction of SCM+FSA; for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
let Ig be good really-closed Program of SCM+FSA; ( ( Ig is parahalting or Ig is_halting_on Initialized s,p ) implies DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s)))) )
set I = Ig;
assume A1:
( Ig is parahalting or Ig is_halting_on Initialized s,p )
; DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
then A2:
for f being FinSeq-Location holds (IExec ((Ig ";" j),p,s)) . f = (Exec (j,(IExec (Ig,p,s)))) . f
by Th11;
for a being Int-Location holds (IExec ((Ig ";" j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a
by A1, Th10;
hence
DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
by A2, SCMFSA_M:2; verum