let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being halt-free parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a
let s be State of SCMPDS; :: thesis: for I being halt-free parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a
let I be halt-free parahalting Program of SCMPDS; :: thesis: for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a
let J be shiftable Program of SCMPDS; :: thesis: for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a
let a be Int_position ; :: thesis: ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a )
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by SCMPDS_6:34, SCMPDS_6:35;
assume ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P ) ; :: thesis: (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a
hence (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a by A1, SCMPDS_7:49; :: thesis: verum