let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
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 = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . f = (IExec (I,P,s)) . f ) )
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 = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . f = (IExec (I,P,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 = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . f = (IExec (I,P,s)) . f ) )
set D = Data-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 = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . f = (IExec (I,P,s)) . f ) ) )
assume A1: s . a > 0 ; :: thesis: ex s2 being State of SCM+FSA ex k being Element of NAT st
( s2 = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),s2,k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),s2,k)) . f = (IExec (I,P,s)) . f ) )
set Is = Initialize (Initialized s);
set Q = while>0 (a,I);
set s1 = Initialized s;
set P1 = P +* I;
take Initialized s ; :: thesis: ex k being Element of NAT st
( Initialized s = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f ) )
set P2 = P +* (while>0 (a,I));
take k = (LifeSpan ((P +* I),(Initialized s))) + 3; :: thesis: ( Initialized s = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 & IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f ) )
thus ( Initialized s = Initialized s & k = (LifeSpan ((P +* I),(Initialized s))) + 3 ) ; :: thesis: ( IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) = 0 & ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f ) )
A3: I is_halting_onInit s,P by SCM_HALT:26;
then A4: P +* I halts_on Initialized s by SCM_HALT:def 5;
Initialized s = Initialize (Initialized s) by MEMSTR_0:44
.= Initialize (Initialized s)
.= Initialize (Initialized s) ;
then P +* I halts_on Initialize (Initialized s) by A4;
then A5: I is_halting_on Initialized s,P by SCMFSA7B:def 7;
A6: I is_closed_onInit s,P by SCM_HALT:25;
hence IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) = 0 by A1, A3, Th21; :: thesis: ( ( for b being Int-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (IExec (I,P,s)) . b ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f ) )
set s4 = Comput ((P +* (while>0 (a,I))),(Initialized s),k);
set s3 = Comput ((P +* I),(Initialized s),(LifeSpan ((P +* I),(Initialized s))));
A7: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;
A9: Comput ((P +* I),(Initialized s),(LifeSpan ((P +* I),(Initialized s)))) = Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s))))) by A7;
A10: DataPart (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) = DataPart (Comput ((P +* I),(Initialized s),(LifeSpan ((P +* I),(Initialized s))))) by A1, A3, A6, Th21;

hereby :: thesis: for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f

let b be Int-Location ; :: thesis: (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (IExec (I,P,s)) . b
thus (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . b = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . b by A9, A10, SCMFSA6A:7
.= (IExec (I,P,s)) . b by A5, Th10 ; :: thesis: verum

end;

hereby :: thesis: verum

let f be FinSeq-Location ; :: thesis: (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (IExec (I,P,s)) . f
thus (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f by A9, A10, SCMFSA6A:7
.= (IExec (I,P,s)) . f by A5, SCMFSA8C:58 ; :: thesis: verum

end;