let n be Ordinal; :: thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT p,T in Upper_Support p,T,i
let T be connected TermOrder of n; :: thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT p,T in Upper_Support p,T,i
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT p,T in Upper_Support p,T,i
let p be non-zero Polynomial of n,L; :: thesis: for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT p,T in Upper_Support p,T,i
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= card (Support p) implies HT p,T in Upper_Support p,T,i )
assume that
A1: 1 <= i and
A2: i <= card (Support p) ; :: thesis: HT p,T in Upper_Support p,T,i
p <> 0_ n,L by POLYNOM7:def 2;
then Support p <> {} by POLYNOM7:1;
then A3: HT p,T in Support p by TERMORD:def 6;
set u = Upper_Support p,T,i;
consider x being Element of Upper_Support p,T,i;
A4: Upper_Support p,T,i <> {} by A1, A2, Def2, CARD_1:47;
then A5: x in Upper_Support p,T,i ;
then reconsider x9 = x as Element of Bags n ;
Upper_Support p,T,i c= Support p by A2, Def2;
then x9 <= HT p,T,T by A5, TERMORD:def 6;
hence HT p,T in Upper_Support p,T,i by A2, A4, A3, Def2; :: thesis: verum