let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let s be State of SCM+FSA; :: thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let a be read-write Int-Location ; :: thesis: ( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A1: I is_halting_on Initialized s,P by SCMFSA7B:25;
for s being State of SCM+FSA st Start-At (0,SCM+FSA) c= s holds
for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st if>0 (a,I,J) c= P holds
P halts_on s

proof

let s be State of SCM+FSA; :: thesis: ( Start-At (0,SCM+FSA) c= s implies for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st if>0 (a,I,J) c= P holds
P halts_on s )
assume Start-At (0,SCM+FSA) c= s ; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st if>0 (a,I,J) c= P holds
P halts_on s
then A3: s = Initialize s by FUNCT_4:104;
let Q be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( if>0 (a,I,J) c= Q implies Q halts_on s )
assume A4: if>0 (a,I,J) c= Q ; :: thesis: Q halts_on s
A5: if>0 (a,I,J) c= Q by A4;
A6: I is_closed_on s,Q by SCMFSA7B:24;
A7: I is_halting_on s,Q by SCMFSA7B:25;
A8: J is_halting_on s,Q by SCMFSA7B:25;
A9: J is_closed_on s,Q by SCMFSA7B:24;
A10: Q +* (if>0 (a,I,J)) = Q by A5, FUNCT_4:104;

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

suppose s . a > 0 ; :: thesis: Q halts_on s

then if>0 (a,I,J) is_halting_on s,Q by A6, A7, Th22;
hence Q halts_on s by A3, A10, SCMFSA7B:def 8; :: thesis: verum

end;

suppose s . a <= 0 ; :: thesis: Q halts_on s

then if>0 (a,I,J) is_halting_on s,Q by A9, A8, Th24;
hence Q halts_on s by A3, A10, SCMFSA7B:def 8; :: thesis: verum

end;

end;

end;

hence if>0 (a,I,J) is parahalting by SCMFSA6B:def 3; :: thesis: ( ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
thus ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A1, Th23, SCMFSA7B:24; :: thesis: ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
J is_halting_on Initialized s,P by SCMFSA7B:25;
hence ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by Th25, SCMFSA7B:24; :: thesis: verum