let I be good parahalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a holds
Initialize ((intloc 0) .--> 1) is Times (a,I) -halted
let a be read-write Int-Location ; :: thesis: ( not I destroys a implies Initialize ((intloc 0) .--> 1) is Times (a,I) -halted )
assume A1: not I destroys a ; :: thesis: Initialize ((intloc 0) .--> 1) is Times (a,I) -halted

now

let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA holds Times (a,I) is_halting_on Initialized b₂,b₃
let P be Instruction-Sequence of SCM+FSA; :: thesis: Times (a,I) is_halting_on Initialized b₁,b₂

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

suppose s . a > 0 ; :: thesis: Times (a,I) is_halting_on Initialized b₁,b₂

then A2: (Initialized 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:22;
(Initialized s) . (intloc 0) = 1 by SCMFSA6A:38;
then Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialized s,P by A1, A2, A3, Th117;
hence Times (a,I) is_halting_on Initialized s,P by A2, Th68; :: thesis: verum

end;

suppose A4: s . a <= 0 ; :: thesis: Times (a,I) is_halting_on Initialized b₁,b₂

A5: Stop SCM+FSA is_closed_on Initialized s,P by SCMFSA7B:18;
A6: Stop SCM+FSA is_halting_on Initialized s,P by SCMFSA7B:19;
(Initialized s) . a <= 0 by A4, SCMFSA6C:3;
hence Times (a,I) is_halting_on Initialized s,P by A5, A6, SCMFSA8B:21; :: thesis: verum

end;

end;

end;

hence Initialize ((intloc 0) .--> 1) is Times (a,I) -halted by Th24; :: thesis: verum