set D = Int-Locations \/ FinSeq-Locations ;
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 (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation 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 (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
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 (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation 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 (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
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 (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
take k = (LifeSpan (s +* (Initialized I))) + 3; :: thesis: ( s2 = s +* (Initialized (while>0 a,I)) & k = (LifeSpan (s +* (Initialized I))) + 3 & IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
thus ( s2 = s +* (Initialized (while>0 a,I)) & k = (LifeSpan (s +* (Initialized I))) + 3 ) ; :: thesis: ( IC (Computation s2,k) = insloc 0 & ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
set Is = (Initialize s) +* (I +* (Start-At (insloc 0 )));
A2: I is_halting_onInit s by SCM_HALT:36;
then s +* (Initialized I) is halting by SCM_HALT:def 5;
then (Initialize s) +* (I +* (Start-At (insloc 0 ))) is halting by SCMFSA8A:13;
then A3: I is_halting_on Initialize s by SCMFSA7B:def 8;
A4: I is_closed_onInit s by SCM_HALT:35;
hence IC (Computation s2,k) = insloc 0 by A1, A2, Th21; :: thesis: ( ( for b being Int-Location holds (Computation s2,k) . b = (IExec I,s) . b ) & ( for f being FinSeq-Location holds (Computation s2,k) . f = (IExec I,s) . f ) )
set s4 = Computation s2,k;
set s3 = Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)));
A5: Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))) = Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (Initialized I))) by SCMFSA8A:13
.= Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* (I +* (Start-At (insloc 0 ))))) by SCMFSA8A:13 ;
A6: DataPart (Computation s2,k) = DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) by A1, A2, A4, Th21;

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

let b be Int-Location ; :: thesis: (Computation s2,k) . b = (IExec I,s) . b
thus (Computation s2,k) . b = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* (I +* (Start-At (insloc 0 )))))) . 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: (Computation s2,k) . f = (IExec I,s) . f
thus (Computation s2,k) . f = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* (I +* (Start-At (insloc 0 )))))) . f by A5, A6, SCMFSA6A:38
.= (IExec I,s) . f by A3, SCMFSA8C:87 ; :: thesis: verum

end;