let s be State of SCM+FSA; :: thesis: for I being parahalting good Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 holds
( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s )
let I be parahalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 holds
( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s )
let a be read-write Int-Location ; :: thesis: ( not I destroys a & s . (intloc 0) = 1 implies ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s ) )
assume A1: not I destroys a ; :: thesis: ( not s . (intloc 0) = 1 or ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s ) )
assume A2: s . (intloc 0) = 1 ; :: thesis: ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s )

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

suppose A3: s . a > 0 ; :: thesis: ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s )

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;
then Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on s by A1, A2, A3, Th117;
hence ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s ) by A3, Th68; :: thesis: verum

end;

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

A5: Stop SCM+FSA is_halting_on s by SCMFSA7B:25;
Stop SCM+FSA is_closed_on s by SCMFSA7B:24;
hence ( Times (a,I) is_closed_on s & Times (a,I) is_halting_on s ) by A4, A5, SCMFSA8B:24; :: thesis: verum

end;

end;