let s be State of ; :: thesis: for I being InitClosed Program of st ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) holds
for J being Program of
for k being Element of NAT st k <= LifeSpan (s +* (Initialized I)) holds
Computation (s +* (Initialized I)),k, Computation (s +* (Initialized (I ';' J))),k equal_outside NAT
let I be InitClosed Program of ; :: thesis: ( ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) implies for J being Program of
for k being Element of NAT st k <= LifeSpan (s +* (Initialized I)) holds
Computation (s +* (Initialized I)),k, Computation (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
for k being Element of NAT st k <= LifeSpan (s +* (Initialized I)) holds
Computation (s +* (Initialized I)),k, Computation (s +* (Initialized (I ';' J))),k equal_outside NAT
set s1 = s +* (Initialized I);
let J be Program of ; :: thesis: for k being Element of NAT st k <= LifeSpan (s +* (Initialized I)) holds
Computation (s +* (Initialized I)),k, Computation (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[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized I)) implies Computation (s +* (Initialized I)),$1, Computation (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 (s +* (Initialized I)) implies Computation (s +* (Initialized I)),m, Computation (s +* (Initialized (I ';' J))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
assume A7: m + 1 <= LifeSpan (s +* (Initialized I)) ; :: thesis: Computation (s +* (Initialized I)),(m + 1), Computation (s +* (Initialized (I ';' J))),(m + 1) equal_outside NAT
then A8: IC (Computation (s +* (Initialized I)),m) = IC (Computation (s +* (Initialized (I ';' J))),m) by A6, AMI_1:121, NAT_1:13;
A9: I ';' J c= Computation (s +* (Initialized (I ';' J))),m by A3, Th13, AMI_1:81;
A10: Computation (s +* (Initialized I)),(m + 1) = Following (Computation (s +* (Initialized I)),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (Initialized I)),m)),(Computation (s +* (Initialized I)),m) ;
A11: Computation (s +* (Initialized (I ';' J))),(m + 1) = Following (Computation (s +* (Initialized (I ';' J))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (Initialized (I ';' J))),m)),(Computation (s +* (Initialized (I ';' J))),m) ;
A12: IC (Computation (s +* (Initialized I)),m) in dom I by A2, Def1;
I c= Computation (s +* (Initialized I)),m by A2, Th13, AMI_1:81;
then A13: CurInstr (Computation (s +* (Initialized I)),m) = I . (IC (Computation (s +* (Initialized I)),m)) by A12, GRFUNC_1:8;
m < LifeSpan (s +* (Initialized I)) by A7, NAT_1:13;
then I . (IC (Computation (s +* (Initialized I)),m)) <> halt SCM+FSA by A1, A13, AMI_1:def 46;
then CurInstr (Computation (s +* (Initialized I)),m) = (I ';' J) . (IC (Computation (s +* (Initialized I)),m)) by A12, A13, SCMFSA6A:54
.= CurInstr (Computation (s +* (Initialized (I ';' J))),m) by A8, A12, A9, A5, GRFUNC_1:8 ;
hence Computation (s +* (Initialized I)),(m + 1), Computation (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 (insloc 0 )))) +* I,s +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) equal_outside NAT & s +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))),(s +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) +* (I ';' J) equal_outside NAT ) by AMI_1:120, FUNCT_7:28;
A15: s +* (Initialized (I ';' J)) = s +* ((I ';' J) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) by FUNCT_4:15
.= (s +* (I ';' J)) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) by FUNCT_4:15
.= (s +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) +* (I ';' J) by Th19 ;
A16: ( Computation (s +* (Initialized I)),0 = s +* (Initialized I) & Computation (s +* (Initialized (I ';' J))),0 = s +* (Initialized (I ';' J)) ) by AMI_1:13;
s +* (Initialized I) = s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) by FUNCT_4:15
.= (s +* I) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) by FUNCT_4:15
.= (s +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) +* 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