let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let s be State of SCMPDS; :: thesis: for I being parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let I be parahalting Program of SCMPDS; :: thesis: for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let J be shiftable Program of SCMPDS; :: thesis: ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) )
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: ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
hence ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) by A1, SCMPDS_7:43; :: thesis: verum