let s be State of SCM+FSA ; :: thesis: for I being paraclosed Program of SCM+FSA st s +* (I +* (Start-At (insloc 0 ))) is halting holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) 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 paraclosed Program of SCM+FSA ; :: thesis: ( s +* (I +* (Start-At (insloc 0 ))) is halting implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),k equal_outside NAT )
assume A1: s +* (I +* (Start-At (insloc 0 ))) is halting ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) 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 SCM+FSA ; :: thesis: for k being Element of NAT st k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) 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 ( $1 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies 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] )
assume A6: ( m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies Computation (s +* (I +* (Start-At (insloc 0 )))),m, Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
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 A7: I ';' J c= Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m by AMI_1:81;
A8: 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) ;
A9: 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) ;
A10: 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 I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
then A11: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A10, GRFUNC_1:8;
assume A12: m + 1 <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1), Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),(m + 1) equal_outside NAT
then A13: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = IC (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by A6, AMI_1:121, NAT_1:13;
m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) by A12, NAT_1:13;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A1, A11, AMI_1:def 46;
then CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (I ';' J) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A10, A11, SCMFSA6A:54
.= CurInstr (Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),m) by A13, A10, A7, A5, GRFUNC_1:8 ;
hence Computation (s +* (I +* (Start-At (insloc 0 )))),(m + 1), Computation (s +* ((I ';' J) +* (Start-At (insloc 0 )))),(m + 1) equal_outside NAT by A6, A12, A8, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

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