let s1, s2 be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1 & I is_halting_on s1 & I +* (Start-At (insloc 0 )) c= s1 & I +* (Start-At (insloc 0 )) c= s2 & ex k being Element of NAT st Computation s1,k,s2 equal_outside NAT holds
Result s1, Result s2 equal_outside NAT
set A = NAT ;
set D = Int-Locations \/ FinSeq-Locations ;
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & I +* (Start-At (insloc 0 )) c= s1 & I +* (Start-At (insloc 0 )) c= s2 & ex k being Element of NAT st Computation s1,k,s2 equal_outside NAT implies Result s1, Result s2 equal_outside NAT )
assume A1: I is_closed_on s1 ; :: thesis: ( not I is_halting_on s1 or not I +* (Start-At (insloc 0 )) c= s1 or not I +* (Start-At (insloc 0 )) c= s2 or for k being Element of NAT holds not Computation s1,k,s2 equal_outside NAT or Result s1, Result s2 equal_outside NAT )
assume A2: I is_halting_on s1 ; :: thesis: ( not I +* (Start-At (insloc 0 )) c= s1 or not I +* (Start-At (insloc 0 )) c= s2 or for k being Element of NAT holds not Computation s1,k,s2 equal_outside NAT or Result s1, Result s2 equal_outside NAT )
assume A3: I +* (Start-At (insloc 0 )) c= s1 ; :: thesis: ( not I +* (Start-At (insloc 0 )) c= s2 or for k being Element of NAT holds not Computation s1,k,s2 equal_outside NAT or Result s1, Result s2 equal_outside NAT )
then A4: s1 = s1 +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:79;
then A5: s1 is halting by A2, SCMFSA7B:def 8;
then consider n being Element of NAT such that
A6: CurInstr (Computation s1,n) = halt SCM+FSA by AMI_1:def 20;
assume I +* (Start-At (insloc 0 )) c= s2 ; :: thesis: ( for k being Element of NAT holds not Computation s1,k,s2 equal_outside NAT or Result s1, Result s2 equal_outside NAT )
then A7: s2 = s2 +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:79;
given k being Element of NAT such that A8: Computation s1,k,s2 equal_outside NAT ; :: thesis: Result s1, Result s2 equal_outside NAT
set s3 = Computation s1,k;
A9: IC SCM+FSA in dom (Computation s1,k) by AMI_1:94;
I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then I c= s1 by A3, XBOOLE_1:1;
then A10: I c= Computation s1,k by AMI_1:86;
IC (Computation s1,k) = IC s2 by A8, SCMFSA8A:6
.= IC ((s2 +* I) +* (Start-At (insloc 0 ))) by A7, FUNCT_4:15
.= insloc 0 by AMI_1:111 ;
then (IC SCM+FSA ) .--> (insloc 0 ) c= Computation s1,k by A9, FUNCOP_1:88;
then I +* (Start-At (insloc 0 )) c= Computation s1,k by A10, FUNCT_4:92;
then A11: Computation s1,k = (Computation s1,k) +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:79;

A12: now

let n be Element of NAT ; :: thesis: IC (Computation (Computation s1,k),n) in dom I
IC (Computation (Computation s1,k),n) = IC (Computation s1,(k + n)) by AMI_1:51;
hence IC (Computation (Computation s1,k),n) in dom I by A1, A4, SCMFSA7B:def 7; :: thesis: verum

end;

CurInstr (Computation (Computation s1,k),n) = CurInstr (Computation s1,(k + n)) by AMI_1:51
.= CurInstr (Computation s1,n) by A6, AMI_1:52, NAT_1:11 ;
then Computation s1,k is halting by A6, AMI_1:def 20;
then A13: I is_halting_on Computation s1,k by A11, SCMFSA7B:def 8;
A14: DataPart (Computation s1,k) = DataPart s2 by A8, SCMFSA8A:6;
consider k being Element of NAT such that
A15: CurInstr (Computation s1,k) = halt SCM+FSA by A5, AMI_1:def 20;
A16: s1 . (IC (Computation s1,k)) = halt SCM+FSA by A15, AMI_1:54;
I is_closed_on Computation s1,k by A11, A12, SCMFSA7B:def 7;
then Result (Computation s1,k), Result s2 equal_outside NAT by A7, A14, A11, A13, Th101;
hence Result s1, Result s2 equal_outside NAT by A16, AMI_1:57; :: thesis: verum