let I be good InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a holds
Initialized (Times (a,I)) is halting
let a be read-write Int-Location ; :: thesis: ( not I destroys a implies Initialized (Times (a,I)) is halting )
assume A1: not I destroys a ; :: thesis: Initialized (Times (a,I)) is halting

now

let s be State of SCM+FSA; :: thesis: Initialized (Times (a,I)) is_halting_on b₁

per cases ( s . a > 0 or s . a <= 0 ) ;

suppose s . a > 0 ; :: thesis: Initialized (Times (a,I)) is_halting_on b₁

then A2: (Initialized s) . a > 0 by SCMFSA6C:3;
( (Initialized s) . (intloc 0) = 1 & Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) = loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) ) by SCMFSA6A:63, SCMFSA6C:3;
then Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialized s by A1, A2, Th73;
then Times (a,I) is_halting_on Initialized s by A2, SCMFSA8C:68;
hence Initialized (Times (a,I)) is_halting_on s by SCMFSA8C:22; :: thesis: verum

end;

suppose A3: s . a <= 0 ; :: thesis: Initialized (Times (a,I)) is_halting_on b₁

A4: ( Stop SCM+FSA is_closed_on Initialized s & Stop SCM+FSA is_halting_on Initialized s ) by SCMFSA7B:24, SCMFSA7B:25;
(Initialized s) . a <= 0 by A3, SCMFSA6C:3;
then Times (a,I) is_halting_on Initialized s by A4, SCMFSA8B:24;
hence Initialized (Times (a,I)) is_halting_on s by SCMFSA8C:22; :: thesis: verum

end;

end;

end;

hence Initialized (Times (a,I)) is halting by SCMFSA8C:24; :: thesis: verum