let P1, P2 be Instruction-Sequence of SCM+FSA; 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; 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 ; ( 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
; 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 verum
let k be
Nat;
( 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;
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
;
verum
end;