let s be State of SCM+FSA; :: thesis: for I being InitClosed Program of SCM+FSA st ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) 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 InitClosed Program of SCM+FSA; :: thesis: ( ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) 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: ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) 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 st k <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) 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 s2 = s +* (Initialized (I ';' J));
A2: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
defpred S₁[ Nat] means ( $1 <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) implies 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] )
assume A6: ( m <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) implies 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]
assume A7: m + 1 <= LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) ; :: thesis: Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1)), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1)) equal_outside NAT
then A8: IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) = IC (Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) by A6, COMPOS_1:24, NAT_1:13;
A9: I ';' J c= Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by A3, Th13, AMI_1:81;
A10: 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;
A11: 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))) ;
A12: 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;
I c= Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by A2, Th13, AMI_1:81;
then A13: 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 A12, Y, GRFUNC_1:8;
Y: (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;
m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) by A7, NAT_1:13;
then I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) <> halt SCM+FSA by A1, A13, EXTPRO_1:def 14;
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 A12, A13, SCMFSA6A:54
.= CurInstr ((ProgramPart (s +* (Initialized (I ';' J)))),(Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m))) by A8, A12, A9, A5, Y, GRFUNC_1:8 ;
hence Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1)), Comput ((ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1)) equal_outside NAT by A6, A7, A10, A11, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

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 ;
A16: ( 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;
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 A17: S₁[ 0 ] by A15, A14, A16, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A17, A4); :: thesis: verum