let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let I be halt-free Program of SCMPDS; :: thesis: for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let J be parahalting shiftable Program of SCMPDS; :: thesis: for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let a be Int_position; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a )
( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P ) by ;
hence ( I is_closed_on s,P & I is_halting_on s,P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a ) by SCMPDS_7:30; :: thesis: verum