Definition:
Subtraction: if
.
Division: if
.
Theorem:
8.
,
.
proof of 8: By F3 ,
;
By F5 ,
;
By F3 ,
;
By Thm P1,
.
9.
s.t.
;
s.t.
.
proof of 9: By F3 ,
.
10.
.
11.
.
12.
.
proof of 12: <= : By P8 , if
, then
;
By P8 , if
, then
.
=> : Assume
, if a = 0 we are done;
Otherwise , by P8 ,
and we have
;
by cancellation (P2) ,
.
.
proof: By F5 ,
Theorem :
s.t.
1.
;
2.
;
3.
.
......
In F2 ,