let s be State of SCM+FSA; :: thesis: for I being keepInt0_1 Program of SCM+FSA st not ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k) equal_outside NAT
let I be keepInt0_1 Program of SCM+FSA; :: thesis: ( not ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) implies for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k) equal_outside NAT )
assume A1: not ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k) equal_outside NAT
set s1 = s +* (Initialized I);
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT holds Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k) equal_outside NAT
A2: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
set s2 = s +* (Initialized (I ';' J));
defpred S₁[ Nat] means Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),$1), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),$1) equal_outside NAT ;
A3: Initialized (I ';' J) c= s +* (Initialized (I ';' J)) by FUNCT_4:26;
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated (J,(card I))))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated (J,(card I))))) by FUNCT_4:105 ;
then A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
set sx = s +* (Initialized (I ';' J));
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
A6: Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1)) = Following ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) ;
T: ProgramPart (s +* (Initialized (I ';' J))) = ProgramPart (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) by AMI_1:123;
A7: Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1)) = Following ((ProgramPart (s +* (Initialized (I ';' J)))),(Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (s +* (Initialized (I ';' J)))),(Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)))),(Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) ;
assume A8: Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) equal_outside NAT ; :: thesis: S₁[m + 1]
then A9: IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) = IC (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) by COMPOS_1:24;
A10: I ';' J c= Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by A3, Th13, AMI_1:81;
A11: IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) in dom I by A2, Def1;
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) by AMI_1:123;
Y: (ProgramPart (s +* (Initialized I))) /. (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by TX, COMPOS_1:38;
Z: (ProgramPart (s +* (Initialized (I ';' J)))) /. (IC (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) = (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) . (IC (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) by T, COMPOS_1:38;
I c= Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by A2, Th13, AMI_1:81;
then A12: CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by A11, Y, GRFUNC_1:8;
then I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) <> halt SCM+FSA by A1, EXTPRO_1:30;
then CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = (I ';' J) . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by A11, A12, SCMFSA6A:54
.= CurInstr ((ProgramPart (s +* (Initialized (I ';' J)))),(Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) by A9, A11, A10, A5, Z, GRFUNC_1:8 ;
hence S₁[m + 1] by A8, A6, A7, SCMFSA6A:32; :: thesis: verum

end;

A13: ( Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),0) = s +* (Initialized I) & Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),0) = s +* (Initialized (I ';' J)) ) by EXTPRO_1:3;
A14: ( (s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) +* I,s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) equal_outside NAT & s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))),(s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A15: s +* (Initialized (I ';' J)) = s +* ((I ';' J) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) by FUNCT_4:15
.= (s +* (I ';' J)) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) by FUNCT_4:15
.= (s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) +* (I ';' J) by Th19 ;
s +* (Initialized I) = s +* (I +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) by FUNCT_4:15
.= (s +* I) +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA))) by FUNCT_4:15
.= (s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))) +* I by Th19 ;
then A16: S₁[ 0 ] by A15, A14, A13, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A16, A4); :: thesis: verum