let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
let s be State of SCM+FSA; :: thesis: for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
let I be good InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a & s . a > 0 holds
ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
let a be read-write Int-Location ; :: thesis: ( not I destroys a & s . a > 0 implies ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) ) )
assume A1: not I destroys a ; :: thesis: ( not s . a > 0 or ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) ) )
reconsider I1 = I ';' (SubFrom (a,(intloc 0))) as InitHalting Program of SCM+FSA ;
set P = if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))));
assume A2: s . a > 0 ; :: thesis: ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
set Is = (Initialized s) +* (Initialize ((intloc 0) .--> 1));
take s2 = s +* (Initialize ((intloc 0) .--> 1)); :: thesis: ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
take p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))); :: thesis: ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
set s1 = s +* (Initialize ((intloc 0) .--> 1));
set p1 = p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))));
take k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1; :: thesis: ( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
thus ( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 ) ; :: thesis: ( (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
A3: (Initialized s) +* (Initialize ((intloc 0) .--> 1)) = Initialized s by FUNCT_4:99
.= s +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:99
.= s +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
.= Initialize (Initialized s) by FUNCT_4:15 ;
A4: I1 is_halting_onInit s,p by Th36;
then A5: I1 is_halting_on Initialized s,p by Th41;
A6: IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),p,s) = (IExec (I1,p,s)) +* (Start-At ((((card (Goto 2)) + (card I1)) + 3),SCM+FSA)) by A2, A4, Th46, Th35;
then A7: (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),p,s)) . a = (IExec (I1,p,s)) . a by SCMFSA_3:11;
( I1 is_closed_onInit s,p & I1 is_halting_onInit s,p ) by Th35, Th36;
then A8: ( if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_onInit s,p & if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_onInit s,p ) by A2, Th45;
Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1)) = Following (p2,(Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1)))))))) by EXTPRO_1:4
.= Exec ((CurInstr (p2,(Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))))))),(Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1)))))))) ;
then A9: Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1)) = Exec ((goto 0),(Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1)))))))) by A8, Th71;
A10: ( I1 is_closed_onInit s,p & I1 is_halting_onInit s,p ) by Th35, Th36;
then if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_onInit s,p by A2, Th45;
then A11: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_on Initialized s,p by Th40;
if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_onInit s,p by A2, A10, Th45;
then A12: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_on Initialized s,p by Th41;
A13: (Initialized s) +* (Initialize ((intloc 0) .--> 1)) = s +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:99;
OO: (Initialized s) +* (Initialize ((intloc 0) .--> 1)) = Initialize (Initialized s) by A3;
XX: NPP (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) = NPP (Comput ((p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) by A8, A13, Th68;

A14: now

let b be Int-Location ; :: thesis: (Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . b = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . b
(Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . b = (Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))))) . b by A9, SCMFSA_2:95;
hence (Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . b = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . b by A13, XX, SCMFSA10:92; :: thesis: verum

end;

then (Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . a = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . a
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),p,s)) . a by A3, A12, SCMFSA8C:87 ;
hence (Comput (p2,s2,k)) . a = (Comput ((p +* I1),(Initialize (Initialized s)),(LifeSpan ((p +* I1),(Initialize (Initialized s)))))) . a by A5, A7, SCMFSA8C:87
.= (s . a) - 1 by A1, Th64 ;
:: thesis: ( (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )
(Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . (intloc 0) = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . (intloc 0) by A14
.= 1 by A12, A11, OO, SCMFSA8C:96 ;
hence (Comput (p2,s2,k)) . (intloc 0) = 1 ; :: thesis: ( ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

hereby :: thesis: ( ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

let b be read-write Int-Location ; :: thesis: ( b <> a implies (Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b )
assume A15: b <> a ; :: thesis: (Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b
thus (Comput (p2,s2,k)) . b = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . b by A14
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),p,s)) . b by A3, A12, SCMFSA8C:87
.= (IExec (I1,p,s)) . b by A6, SCMFSA_3:11
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,s)))) . b by Th33
.= (IExec (I,p,s)) . b by A15, SCMFSA_2:91 ; :: thesis: verum

end;

hereby :: thesis: ( IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

let f be FinSeq-Location ; :: thesis: (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f
(Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) . f = (Comput (p2,s2,(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))))) . f by A9, SCMFSA_2:95;
hence (Comput (p2,s2,k)) . f = (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))))) . f by A13, XX, SCMFSA10:93
.= (IExec ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))),p,s)) . f by A3, A12, SCMFSA8C:87
.= (IExec (I1,p,s)) . f by A6, SCMFSA_3:12
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,s)))) . f by Th34
.= (IExec (I,p,s)) . f by SCMFSA_2:91 ;
:: thesis: verum

end;

thus IC (Comput (p2,s2,k)) = 0 by A9, SCMFSA_2:95; :: thesis: for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))
A16: IC (Comput (p2,s2,((LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = 0 by A9, SCMFSA_2:95;

hereby :: thesis: verum

let n be Element of NAT ; :: thesis: ( n <= k implies IC (Comput (p2,s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) )
assume A17: n <= k ; :: thesis: IC (Comput (p2,s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

per cases ( n <= LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1)))) or n = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 ) by A17, NAT_1:8;

suppose A18: n <= LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1)))) ; :: thesis: IC (Comput (p2,s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

A19: ( I1 is_closed_onInit s,p & I1 is_halting_onInit s,p ) by Th35, Th36;
then A20: if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_closed_onInit s,p by A2, Th45;
if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))) is_halting_onInit s,p by A2, A19, Th45;
then NPP (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))),n)) = NPP (Comput (p2,s2,n)) by A18, A20, Th68;
then A21: IC (Comput (p2,s2,n)) = IC (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))),n)) by SCMFSA8A:6;
IC (Comput ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))),n)) in dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by A20, Def4;
hence IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A21, FUNCT_4:105; :: thesis: verum

end;

suppose A22: n = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 ; :: thesis: IC (Comput (p2,s2,b₁)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))

A23: card (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) = ((card (Goto 2)) + (card I1)) + 4 by SCMFSA8B:14
.= ((card I1) + 1) + 4 by SCMFSA8A:29
.= ((card I1) + 3) + 2 ;
card (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) = card (dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))) by CARD_1:104
.= card (dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by FUNCT_4:105
.= card (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by CARD_1:104 ;
hence IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A16, A22, A23, AFINSQ_1:70; :: thesis: verum

end;

end;

end;