Table 2: The lemmas proving Theorem 19.

Name of lemmaDescription in HOL4

DIVISION_LE_SUC . division (a, b) d ==> n. d n <= d (SUC n)
DIVISION_MONO_LE . division (a, b) d ==> m n. m <= n ==> d m <= d n
DIVISION_MONO_LE_SUC . division (a, b) d ==> n. d n <= d (SUC n)
DIVISION_INTERMEDIATE . division (a, b) d   a <= c   c <= b ==>
n. n <= dsize d   d n <= c   c <= d (SUC n)
DIVISION_DSIZE_LE . division (a, b) d (d (SUC n) = d n) ==> dsize d <= n
DIVISION_DSIZE_GE . division (a, b) d   d n < d (SUC n) ==> SUC n <= dsize d
DIVISION_DSIZE_EQ . division (a, b) d   d n < d (SUC n) (d (SUC (SUC n)) = d (SUC n)) ==>
 (dsize d = SUC n)
DIVISION_DSIZE_EQ_ALT . division (a, b) d (d (SUC n) = d n) (i. i < n ==> d i < d (SUC i))
==> (dsize d = n)