let s be State of SCM+FSA; :: thesis: for I being InitHalting Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 holds
ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
let I be InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st s . a > 0 holds
ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
set D = Int-Locations \/ FinSeq-Locations;
let a be read-write Int-Location ; :: thesis: ( s . a > 0 implies ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) ) )
assume A1: s . a > 0 ; :: thesis: ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
set Is = (Initialized s) +* (I +* (Start-At (0,SCM+FSA)));
set P = while>0 (a,I);
set s1 = s +* (Initialized I);
take s2 = s +* (Initialized (while>0 (a,I))); :: thesis: ex k being Element of NAT st
( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
take k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3; :: thesis: ( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 & IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
thus ( s2 = s +* (Initialized (while>0 (a,I))) & k = (LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) + 3 ) ; :: thesis: ( IC (Comput ((ProgramPart s2),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
A2: I is_halting_onInit s by SCM_HALT:36;
then X: ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) by SCM_HALT:def 5;
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
then A3: I is_halting_on Initialized s by X, SCMFSA7B:def 8;
A4: I is_closed_onInit s by SCM_HALT:35;
hence IC (Comput ((ProgramPart s2),s2,k)) = 0 by A1, A2, Th21; :: thesis: ( ( for b being Int-Location holds (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f ) )
set s4 = Comput ((ProgramPart s2),s2,k);
set s3 = Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))));
X: s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
XX: s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
A5: Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) = Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by X
.= Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))))) by XX ;
A6: DataPart (Comput ((ProgramPart s2),s2,k)) = DataPart (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))))) by A1, A2, A4, Th21;

hereby :: thesis: for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f

let b be Int-Location ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,s)) . b
thus (Comput ((ProgramPart s2),s2,k)) . b = (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))))))) . b by A5, A6, SCMFSA6A:38
.= (IExec (I,s)) . b by A3, Th10 ; :: thesis: verum

end;

hereby :: thesis: verum

let f be FinSeq-Location ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,s)) . f
thus (Comput ((ProgramPart s2),s2,k)) . f = (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))))))) . f by A5, A6, SCMFSA6A:38
.= (IExec (I,s)) . f by A3, SCMFSA8C:87 ; :: thesis: verum

end;