let s1, s2 be State of SCM+FSA ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
let I be Program of SCM+FSA ; :: thesis:
assume A1: I is_closed_on s1 ; :: thesis:
set ss2 = s2 +* (I +* (Start-At 0 ,SCM+FSA ));
set ss1 = s1 +* (I +* (Start-At 0 ,SCM+FSA ));
assume A2: I is_halting_on s1 ; :: thesis:
assume A3: DataPart s1 = DataPart s2 ; :: thesis:
A4: I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
A5: I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
I +* (Start-At 0 ,SCM+FSA ) c= s2 +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then A6: I c= s2 +* (I +* (Start-At 0 ,SCM+FSA )) by A4, XBOOLE_1:1;
I +* (Start-At 0 ,SCM+FSA ) c= s1 +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then A7: I c= s1 +* (I +* (Start-At 0 ,SCM+FSA )) by A5, XBOOLE_1:1;

hereby :: thesis:

let k be Element of NAT ; :: thesis:
A8: IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I by A1, SCMFSA7B:def 7;
I is_closed_on s2 by A1, A2, A3, SCMFSA8B:8;
then A9: for m being Element of NAT st m < k holds
IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),m) in dom I by SCMFSA7B:def 7;
s1 +* (I +* (Start-At 0 ,SCM+FSA )),s2 +* (I +* (Start-At 0 ,SCM+FSA )) equal_outside NAT by A3, SCMFSA8B:7;
hence Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k, Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k equal_outside NAT by A7, A6, A9, SCMFSA6B:21; :: thesis:
then A10: IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k) = IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k) by AMI_1:121;
I is_closed_on s2 by A1, A2, A3, SCMFSA8B:8;
then A11: IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I by SCMFSA7B:def 7;
Y: (ProgramPart (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
thus CurInstr (ProgramPart (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k) = (s2 +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by Y, AMI_1:54
.= I . (IC (Comput (ProgramPart (s2 +* (I +* (Start-At 0 ,SCM+FSA )))),(s2 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A6, A11, GRFUNC_1:8
.= (s1 +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A7, A10, A8, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),k) by Z, AMI_1:54 ; :: thesis:

end;