let s be State of SCM+FSA ; :: thesis: for I being keeping_0 Program of SCM+FSA st not s +* (I +* (Start-At (insloc 0 ))) is halting holds
for J being Program of SCM+FSA
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 SCM+FSA ; :: thesis: ( not s +* (I +* (Start-At (insloc 0 ))) is halting implies for J being Program of SCM+FSA
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 s +* (I +* (Start-At (insloc 0 ))) is halting ; :: thesis: for J being Program of SCM+FSA
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
let J be Program of SCM+FSA ; :: 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
set s1 = s +* (I +* (Start-At (insloc 0 )));
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 )));
A3: (I ';' J) +* (Start-At (insloc 0 )) c= s +* ((I ';' J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
A4: s +* (I +* (Start-At (insloc 0 ))) = (s +* I) +* (Start-At (insloc 0 )) by FUNCT_4:15
.= (s +* (Start-At (insloc 0 ))) +* I by Th14 ;
A5: 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 ;
A6: (s +* (Start-At (insloc 0 ))) +* I,s +* (Start-At (insloc 0 )) equal_outside NAT by AMI_1:120, FUNCT_7:28;
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 ;
A7: s +* (Start-At (insloc 0 )),(s +* (Start-At (insloc 0 ))) +* (I ';' J) equal_outside NAT by AMI_1:120;
( 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;
then A8: S₁[ 0 ] by A4, A5, A6, A7, FUNCT_7:29;
A9: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
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]
A11: 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) ;
A12: 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) ;
A13: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = IC (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by A10, AMI_1:121;
A14: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom I by A2, Def2;
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 A15: I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
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 A16: I ';' J c= Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m by AMI_1:81;
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 A17: dom I c= dom (I ';' J) by XBOOLE_1:7;
A18: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A14, A15, GRFUNC_1:8;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A1, AMI_1:def 20;
then CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (I ';' J) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A14, A18, SCMFSA6A:54
.= CurInstr (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by A13, A14, A16, A17, GRFUNC_1:8 ;
hence S₁[m + 1] by A10, A11, A12, SCMFSA6A:32; :: thesis: verum

end;

thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A8, A9); :: thesis: verum