let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s being 0 -started State of SCM+FSA
for I being really-closed parahalting Program of st I c= P1 & I c= P2 holds
for k being Nat holds
( Comput (P1,s,k) = Comput (P2,s,k) & CurInstr (P1,(Comput (P1,s,k))) = CurInstr (P2,(Comput (P2,s,k))) )
let s be 0 -started State of SCM+FSA; :: thesis: for I being really-closed parahalting Program of st I c= P1 & I c= P2 holds
for k being Nat holds
( Comput (P1,s,k) = Comput (P2,s,k) & CurInstr (P1,(Comput (P1,s,k))) = CurInstr (P2,(Comput (P2,s,k))) )
let I be really-closed parahalting Program of ; :: thesis: ( I c= P1 & I c= P2 implies for k being Nat holds
( Comput (P1,s,k) = Comput (P2,s,k) & CurInstr (P1,(Comput (P1,s,k))) = CurInstr (P2,(Comput (P2,s,k))) ) )
assume that
A1: I c= P1 and
A2: I c= P2 ; :: thesis: for k being Nat holds
( Comput (P1,s,k) = Comput (P2,s,k) & CurInstr (P1,(Comput (P1,s,k))) = CurInstr (P2,(Comput (P2,s,k))) )

hereby :: thesis: verum

let k be Nat; :: thesis: ( Comput (P1,s,k) = Comput (P2,s,k) & CurInstr (P2,(Comput (P2,s,k))) = CurInstr (P1,(Comput (P1,s,k))) )
IC s = 0 by MEMSTR_0:def 11;
then A3: IC s in dom I by AFINSQ_1:65;
then A4: IC (Comput (P1,s,k)) in dom I by A1, AMISTD_1:21;
A5: IC (Comput (P2,s,k)) in dom I by A2, A3, AMISTD_1:21;
for m being Nat st m < k holds
IC (Comput (P2,s,m)) in dom I by A2, AMISTD_1:21, A3;
hence A6: Comput (P1,s,k) = Comput (P2,s,k) by A1, A2, AMISTD_2:10; :: thesis: CurInstr (P2,(Comput (P2,s,k))) = CurInstr (P1,(Comput (P1,s,k)))
thus CurInstr (P2,(Comput (P2,s,k))) = P2 . (IC (Comput (P2,s,k))) by PBOOLE:143
.= I . (IC (Comput (P2,s,k))) by A5, A2, GRFUNC_1:2
.= P1 . (IC (Comput (P1,s,k))) by A6, A4, A1, GRFUNC_1:2
.= CurInstr (P1,(Comput (P1,s,k))) by PBOOLE:143 ; :: thesis: verum

end;