let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis:
let s1, s2 be State of SCM+FSA; :: thesis:
let I be InitHalting Program of SCM+FSA; :: thesis:
assume that
A1: Initialize ((intloc 0) .--> 1) c= s1 and
A2: Initialize ((intloc 0) .--> 1) c= s2 and
A3: I c= P1 and
A4: I c= P2 and
A5: s1 = s2 ; :: thesis:

hereby :: thesis:

let k be Element of NAT ; :: thesis:
A6: IC (Comput (P1,s1,k)) in dom I by A1, A3, SCM_HALT:def 1;
A7: IC (Comput (P2,s2,k)) in dom I by A2, A4, SCM_HALT:def 1;
for m being Element of NAT st m < k holds
IC (Comput (P2,s2,m)) in dom I by A2, A4, SCM_HALT:def 1;
hence Comput (P1,s1,k) = Comput (P2,s2,k) by A5, A3, A4, AMISTD_2:10; :: thesis:
then A8: IC (Comput (P1,s1,k)) = IC (Comput (P2,s2,k)) ;
thus CurInstr (P2,(Comput (P2,s2,k))) = P2 . (IC (Comput (P2,s2,k))) by PBOOLE:143
.= I . (IC (Comput (P2,s2,k))) by A7, A4, GRFUNC_1:2
.= P1 . (IC (Comput (P1,s1,k))) by A8, A6, A3, GRFUNC_1:2
.= CurInstr (P1,(Comput (P1,s1,k))) by PBOOLE:143 ; :: thesis:

end;