let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_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; :: thesis: for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) holds
DataPart (IExec ((Ig ';' j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
let Ig be good Program of SCM+FSA; :: thesis: for j being parahalting Instruction of SCM+FSA st ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) holds
DataPart (IExec ((Ig ';' j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))
let j be parahalting Instruction of SCM+FSA; :: thesis: ( ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_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 & Ig is_closed_on Initialized s,p ) ) ; :: thesis: 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 Th13;
for a being Int-Location holds (IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a by A1, Th12;
hence DataPart (IExec ((Ig ';' j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s)))) by A2, SCMFSA6A:7; :: thesis: verum