let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being keepInt0_1 Program of SCM+FSA st not p +* I halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT 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 keepInt0_1 Program of SCM+FSA st not p +* I halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT
let I be keepInt0_1 Program of SCM+FSA; :: thesis: ( not p +* I halts_on s +* (Initialized I) implies for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((p +* I),(s +* (Initialized I)),k), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),k) equal_outside NAT )
assume A1: not p +* I halts_on s +* (Initialized I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT 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;
A3: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT 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);
A4: I ';' J c= p +* (I ';' J) by FUNCT_4:26;
defpred S₁[ Nat] means Comput ((p +* I),(s +* (Initialized I)),$1), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),$1) equal_outside NAT ;
A5: 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 A6: dom I c= dom (I ';' J) by XBOOLE_1:7;
set sx = s +* (Initialized (I ';' J));
set px = p +* (I ';' J);
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
A7: 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))) ;
A8: 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))) ;
assume A9: Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m) equal_outside NAT ; :: thesis: S₁[m + 1]
then A10: IC (Comput ((p +* I),(s +* (Initialized I)),m)) = IC (Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m)) by COMPOS_1:24;
A11: IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom I by Def1, A2, A3;
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: (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;
A14: 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;
then I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) <> halt SCM+FSA by A1, EXTPRO_1:30;
then CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) = (I ';' J) . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by A11, A14, SCMFSA6A:54
.= CurInstr ((p +* (I ';' J)),(Comput ((p +* (I ';' J)),(s +* (Initialized (I ';' J))),m))) by A10, A11, A6, A13, GRFUNC_1:8, A4 ;
hence S₁[m + 1] by A9, A7, A8, AMISTD_2:def 20; :: thesis: verum

end;

A15: ( 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;
A16: ( (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;
A17: 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 ;
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 A17, A16, A15, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A5); :: thesis: verum