let n be Ordinal; for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible well-unital distributive add-associative right_zeroed associative commutative doubleLoopStr
for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let T be connected admissible TermOrder of n; for L being non empty non degenerated right_complementable almost_left_invertible well-unital distributive add-associative right_zeroed associative commutative doubleLoopStr
for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let L be non empty non degenerated right_complementable almost_left_invertible well-unital distributive add-associative right_zeroed associative commutative doubleLoopStr ; for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let f, p be non-zero Polynomial of n,L; ( f is_reducible_wrt p,T implies HT (p,T) <= HT (f,T),T )
assume
f is_reducible_wrt p,T
; HT (p,T) <= HT (f,T),T
then consider b being bag of n such that
A1:
( b in Support f & HT (p,T) divides b )
by POLYRED:36;
( b <= HT (f,T),T & HT (p,T) <= b,T )
by A1, TERMORD:10, TERMORD:def 6;
hence
HT (p,T) <= HT (f,T),T
by TERMORD:8; verum