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 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
DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s))))
let s be State of SCM+FSA; :: thesis: for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA 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
DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s))))
let J be Program of SCM+FSA; :: thesis: for Ig being good Program of SCM+FSA 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
DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s))))
let Ig be good Program of SCM+FSA; :: 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 DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s)))) )
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: DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s))))
A3: for f being FinSeq-Location holds (IExec ((Ig ';' J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f by A1, A2, Th9;
for a being Int-Location holds (IExec ((Ig ';' J),p,s)) . a = (IExec (J,p,(IExec (Ig,p,s)))) . a by A1, A2, Th8;
hence DataPart (IExec ((Ig ';' J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s)))) by A3, SCMFSA6A:7; :: thesis: verum