let I be good InitHalting 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 :: thesis: for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA holds Times (a,I) is_halting_on Initialized s,p

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 SCMFSA_M:37;
( (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:22, SCMFSA_M:9;
then Directed (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialized s,p by A1, A2, Th60;
hence Times (a,I) is_halting_on Initialized s,p by A2, SCMFSA8C:39; :: thesis: verum

end;

suppose A3: s . a <= 0 ; :: thesis: Times (a,I) is_halting_on Initialized b₁,b₂

A4: ( Stop SCM+FSA is_closed_on Initialized s,p & Stop SCM+FSA is_halting_on Initialized s,p ) by SCMFSA7B:18, SCMFSA7B:19;
(Initialized s) . a <= 0 by A3, SCMFSA_M:37;
hence Times (a,I) is_halting_on Initialized s,p by A4, SCMFSA8B:21; :: thesis: verum

end;

end;

end;

hence Initialize ((intloc 0) .--> 1) is Times (a,I) -halted by SCMFSA8C:6; :: thesis: verum