let P1, P2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis:
let s1, s2 be State of SCM+FSA; :: thesis:
let I be InitHalting Program of SCM+FSA; :: thesis:
assume that
A1: Initialized I c= s1 and
A2: Initialized I c= s2 and
A3: I c= P1 and
A4: I c= P2 and
A5: s1,s2 equal_outside NAT ; :: thesis:

hereby :: thesis:

let k be Element of NAT ; :: thesis:
A6: IC (Comput (P1,s1,k)) in dom I by A1, SCM_HALT:def 1, A3;
A7: IC (Comput (P2,s2,k)) in dom I by A2, SCM_HALT:def 1, A4;
for m being Element of NAT st m < k holds
IC (Comput (P2,s2,m)) in dom I by A2, SCM_HALT:def 1, A4;
hence Comput (P1,s1,k), Comput (P2,s2,k) equal_outside NAT by A5, AMISTD_2:66, A3, A4; :: thesis:
then A8: IC (Comput (P1,s1,k)) = IC (Comput (P2,s2,k)) by COMPOS_1:24;
thus CurInstr (P2,(Comput (P2,s2,k))) = P2 . (IC (Comput (P2,s2,k))) by PBOOLE:158
.= I . (IC (Comput (P2,s2,k))) by A7, GRFUNC_1:8, A4
.= P1 . (IC (Comput (P1,s1,k))) by A8, A6, GRFUNC_1:8, A3
.= CurInstr (P1,(Comput (P1,s1,k))) by PBOOLE:158 ; :: thesis:

end;