let n be Ordinal; for T being connected TermOrder of n
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
let T be connected TermOrder of n; for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; for p being Polynomial of n,L holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
let p be Polynomial of n,L; ( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
set u = Up (p,T,0);
set l = Low (p,T,0);
A1:
0 <= card (Support p)
;
then
Support (Up (p,T,0)) = Upper_Support (p,T,0)
by Lm3;
then
card (Support (Up (p,T,0))) = 0
by A1, Def2;
then
Support (Up (p,T,0)) = {}
;
hence
Up (p,T,0) = 0_ (n,L)
by POLYNOM7:1; Low (p,T,0) = p
then
(0_ (n,L)) + (Low (p,T,0)) = p
by A1, Th33;
hence
Low (p,T,0) = p
by POLYRED:2; verum