let s be State of ; :: thesis: for I being keeping_0 Program of st not ProgramPart (s +* (I +* (Start-At (insloc 0 )))) halts_on s +* (I +* (Start-At (insloc 0 ))) holds
for J being Program of
for k being Element of NAT holds Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),k equal_outside NAT
set SA0 = Start-At (insloc 0 );
let I be keeping_0 Program of ; :: thesis: ( not ProgramPart (s +* (I +* (Start-At (insloc 0 )))) halts_on s +* (I +* (Start-At (insloc 0 ))) implies for J being Program of
for k being Element of NAT holds Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),k equal_outside NAT )
assume A1: not ProgramPart (s +* (I +* (Start-At (insloc 0 )))) halts_on s +* (I +* (Start-At (insloc 0 ))) ; :: thesis: for J being Program of
for k being Element of NAT holds Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),k equal_outside NAT
set s1 = s +* (I +* (Start-At (insloc 0 )));
let J be Program of ; :: thesis: for k being Element of NAT holds Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),k equal_outside NAT
A2: I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:26;
set s2 = s +* ((I ';' J) +* (Start-At (insloc 0 )));
defpred S₁[ Element of NAT ] means Computation (s +* (I +* (Start-At (insloc 0 )))),$1, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),$1 equal_outside NAT ;
A3: (I ';' J) +* (Start-At (insloc 0 )) c= s +* ((I ';' J) +* (Start-At (insloc 0 ))) 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;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
A6: Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1) = Following (Computation (s +* (I +* (Start-At (insloc 0 )))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m)),(Computation (s +* (I +* (Start-At (insloc 0 )))),m) ;
A7: Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),(m + 1) = Following (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m)),(Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) ;
dom (I ';' J) misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then I ';' J c= (I ';' J) +* (Start-At (insloc 0 )) by FUNCT_4:33;
then I ';' J c= s +* ((I ';' J) +* (Start-At (insloc 0 ))) by A3, XBOOLE_1:1;
then A8: I ';' J c= Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m by AMI_1:81;
A9: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom I by A2, Def2;
assume A10: Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m equal_outside NAT ; :: thesis: S₁[m + 1]
then A11: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = IC (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by AMI_1:121;
dom I misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then I c= I +* (Start-At (insloc 0 )) by FUNCT_4:33;
then I c= s +* (I +* (Start-At (insloc 0 ))) by A2, XBOOLE_1:1;
then I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
then A12: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A9, GRFUNC_1:8;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A1, AMI_1:146;
then CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (I ';' J) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A9, A12, SCMFSA6A:54
.= CurInstr (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by A11, A9, A8, A5, GRFUNC_1:8 ;
hence S₁[m + 1] by A10, A6, A7, SCMFSA6A:32; :: thesis: verum

end;

A13: ( Computation (s +* (I +* (Start-At (insloc 0 )))),0 = s +* (I +* (Start-At (insloc 0 ))) & Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),0 = s +* ((I ';' J) +* (Start-At (insloc 0 ))) ) by AMI_1:13;
A14: ( (s +* (Start-At (insloc 0 ))) +* I,s +* (Start-At (insloc 0 )) equal_outside NAT & s +* (Start-At (insloc 0 )),(s +* (Start-At (insloc 0 ))) +* (I ';' J) equal_outside NAT ) by AMI_1:120, FUNCT_7:28;
A15: s +* ((I ';' J) +* (Start-At (insloc 0 ))) = (s +* (I ';' J)) +* (Start-At (insloc 0 )) by FUNCT_4:15
.= (s +* (Start-At (insloc 0 ))) +* (I ';' J) by Th14 ;
s +* (I +* (Start-At (insloc 0 ))) = (s +* I) +* (Start-At (insloc 0 )) by FUNCT_4:15
.= (s +* (Start-At (insloc 0 ))) +* I by Th14 ;
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