let P be Instruction-Sequence of SCMPDS; 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; 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; 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; 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; ( 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 SCMPDS_6:20, SCMPDS_6:21;
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; verum