
Section 4.1 Interval sets and arithmetic 93
(c) If fR, N , C, , =, g is an ordered division ringoid, then fIR,
e
N ,
+
,
,
/
,
g with
e
N :DfA 2 IR j A \ N ¤ ¿g is also an ordered division ringoid.
Moreover, for all A D Œa
1
, a
2
2 IR and B D Œb
1
, b
2
2 IR n
e
N we have
(E) A 0 ^ 0 <b
1
b
2
) A
/
B D Œa
1
=b
2
, a
2
=b
1
,
(F) A 0 ^ b
1
b
2
< 0 ) A
/
B D Œa
2
=b
2
, a
1
=b
1
,
(G) A 0 ^ 0 <b
1
b
2
) A
/
B D Œa
1
=b
1
, a
2
=b
2
,
(H) A 0 ^ b
1
b
2
< 0 ) A
/
B D Œa
2
=b
1
, a
1
=b
2
.
Proof. (a) Theorem 3.5 directly implies the properties (D1, 2, 3, 4, 5, 7, 8, 9) and (OD5).
It remains to show (A), (D6), (OD1), and (OD2). Let be A D Œa
1
, a
2
, B D Œb
1
, b
2
,
C D Œc
1
, c
2
.
(A): We demonstrate the