let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitClosed Program of SCM+FSA st p +* I halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being InitClosed Program of SCM+FSA st p +* I halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT
let I be InitClosed Program of SCM+FSA; :: thesis: ( p +* I halts_on s +* (Initialized I) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT )
assume A1: p +* I halts_on s +* (Initialized I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT
set s1 = s +* (Initialized I);
set p1 = p +* I;
A2: I c= p +* I by FUNCT_4:26;
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT
set s2 = s +* (Initialized (I ';' J));
set p2 = p +* (I ';' J);
defpred S₁[ Nat] means ( $1 <= LifeSpan ((p +* I),(s +* (Initialized I))) implies Comput ((p +* I),(s +* (Initialized I)),$1), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),$1) equal_outside NAT );
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:105 ;
then A4: dom I c= dom (I ';' J) by XBOOLE_1:7;
set sx = s +* (Initialized (I ';' J));
set px = p +* (I ';' J);
A5: I ';' J c= p +* (I ';' J) by FUNCT_4:26;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan ((p +* I),(s +* (Initialized I))) implies Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
assume A7: m + 1 <= LifeSpan ((p +* I),(s +* (Initialized I))) ; :: thesis: Comput ((p +* I),(s +* (Initialized I)),(m + 1)), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),(m + 1)) equal_outside NAT
then A8: IC (Comput ((p +* I),(s +* (Initialized I)),m)) = IC (Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m)) by A6, COMPOS_1:24, NAT_1:13;
A9: Comput ((p +* I),(s +* (Initialized I)),(m + 1)) = Following ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m)))),(Comput ((p +* I),(s +* (Initialized I)),m))) ;
A10: Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),(m + 1)) = Following ((p +* (I ';' J)),(Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* (I ';' J)),(Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m)))),(Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) ;
Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then A11: IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom I by Def1, A2;
A12: (p +* I) /. (IC (Comput ((p +* I),(s +* (Initialized I)),m))) = (p +* I) . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by PBOOLE:158;
A13: CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) = I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by A11, A12, GRFUNC_1:8, A2;
A14: (p +* (I ';' J)) /. (IC (Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) = (p +* (I ';' J)) . (IC (Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) by PBOOLE:158;
m < LifeSpan ((p +* I),(s +* (Initialized I))) by A7, NAT_1:13;
then I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) <> halt SCM+FSA by A1, A13, EXTPRO_1:def 14;
then CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) = (I ';' J) . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by A11, A13, SCMFSA6A:54
.= (p +* (I ';' J)) . (IC (Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) by A8, A11, A4, GRFUNC_1:8, A5
.= CurInstr ((p +* (I ';' J)),(Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) by A14 ;
hence Comput ((p +* I),(s +* (Initialized I)),(m + 1)), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),(m + 1)) equal_outside NAT by A6, A7, A9, A10, NAT_1:13, AMISTD_2:def 20; :: thesis: verum

end;

A15: ( (s +* (Initialize ((intloc 0) .--> 1))) +* I,s +* (Initialize ((intloc 0) .--> 1)) equal_outside NAT & s +* (Initialize ((intloc 0) .--> 1)),(s +* (Initialize ((intloc 0) .--> 1))) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A16: s +* (Initialized (I ';' J)) = s +* ((I ';' J) +* (Initialize ((intloc 0) .--> 1))) by FUNCT_4:15
.= (s +* (I ';' J)) +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:15
.= (s +* (Initialize ((intloc 0) .--> 1))) +* (I ';' J) by Th19 ;
A17: ( Comput ((p +* I),(s +* (Initialized I)),0) = s +* (Initialized I) & Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),0) = s +* (Initialized (I ';' J)) ) by EXTPRO_1:3;
s +* (Initialized I) = s +* (I +* (Initialize ((intloc 0) .--> 1))) by FUNCT_4:15
.= (s +* I) +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:15
.= (s +* (Initialize ((intloc 0) .--> 1))) +* I by Th19 ;
then A18: S₁[ 0 ] by A16, A15, A17, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A3); :: thesis: verum