let a be Int_position ; :: thesis: for s being State of SCMPDS
for I being halt-free Program of SCMPDS
for J being shiftable Program of SCMPDS st I is_closed_on s & I is_halting_on s & J is_closed_on IExec I,s & J is_halting_on IExec I,s holds
(IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a
let s be State of SCMPDS ; :: thesis: for I being halt-free Program of SCMPDS
for J being shiftable Program of SCMPDS st I is_closed_on s & I is_halting_on s & J is_closed_on IExec I,s & J is_halting_on IExec I,s holds
(IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a
let I be halt-free Program of SCMPDS ; :: thesis: for J being shiftable Program of SCMPDS st I is_closed_on s & I is_halting_on s & J is_closed_on IExec I,s & J is_halting_on IExec I,s holds
(IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a
let J be shiftable Program of SCMPDS ; :: thesis: ( I is_closed_on s & I is_halting_on s & J is_closed_on IExec I,s & J is_halting_on IExec I,s implies (IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a )
assume that
A1: I is_closed_on s and
A2: I is_halting_on s and
A3: J is_closed_on IExec I,s and
A4: J is_halting_on IExec I,s ; :: thesis: (IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a
A5: not a in dom (Start-At ((IC (IExec J,(IExec I,s))) + (card I)),SCMPDS ) by SCMPDS_4:59;
IExec (I ';' J),s = (IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)),SCMPDS ) by A1, A2, A3, A4, Th48;
hence (IExec (I ';' J),s) . a = (IExec J,(IExec I,s)) . a by A5, FUNCT_4:12; :: thesis: verum