let I be parahalting good Program of SCM+FSA ; :: thesis: for a being read-write Int-Location st I does_not_destroy a holds
Initialized (Times a,I) is halting
let a be read-write Int-Location ; :: thesis: ( I does_not_destroy a implies Initialized (Times a,I) is halting )
assume A1: I does_not_destroy 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: (Initialize s) . a > 0 by SCMFSA6C:3;
A3: 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;
(Initialize s) . (intloc 0 ) = 1 by SCMFSA6C:3;
then Directed (loop (if=0 a,(Goto 2),(I ';' (SubFrom a,(intloc 0 ))))) is_pseudo-closed_on Initialize s by A1, A2, A3, Th117;
then Times a,I is_halting_on Initialize s by A2, Th68;
hence Initialized (Times a,I) is_halting_on s by Th22; :: thesis: verum

end;

suppose A4: s . a <= 0 ; :: thesis: Initialized (Times a,I) is_halting_on b₁

A5: Stop SCM+FSA is_closed_on Initialize s by SCMFSA7B:24;
A6: Stop SCM+FSA is_halting_on Initialize s by SCMFSA7B:25;
(Initialize s) . a <= 0 by A4, SCMFSA6C:3;
then Times a,I is_halting_on Initialize s by A5, A6, SCMFSA8B:24;
hence Initialized (Times a,I) is_halting_on s by Th22; :: thesis: verum

end;

end;

end;

hence Initialized (Times a,I) is halting by Th24; :: thesis: verum