Definition:
Subtract: if
belong to
.
Divition: if
belong to F ,
to the power
.
Theorem:
8. For every
belongs to F ,
.
proof of 8: By F3 ,
;
By F5 ,
;
By F3 ,
;
By Thm P1 ,
.
9. There not exists
belongs to F s.t.
;
For every
belongs to F s.t.
is not equal to
.
proof of 9: By F3 ,
is not equal to
.
10.
.
11.
.
12.
.
proof of 12: <= : By P8 , if
, then
;
By P8 , if
, then
.
=> : Assume
, if a = 0 we have done;
Otherwise , by P8 ,
is not equal to
and we have
;
by cancellation (P2) ,
.
.
proof: By F5 ,
Theorem :
There exists !(unique)
s.t.
1.
;
2. For every
belong to Z ,
;
3. >For every
belong to Z ,
.
iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......
In F2 ,