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 1;

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);

set x = the Element of Upper_Support (p,T,i);

A4: Upper_Support (p,T,i) <> {} by A1, A2, Def2, CARD_1:27;

then A5: the Element of Upper_Support (p,T,i) in Upper_Support (p,T,i) ;

then reconsider x9 = the Element of Upper_Support (p,T,i) 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

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 1;

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);

set x = the Element of Upper_Support (p,T,i);

A4: Upper_Support (p,T,i) <> {} by A1, A2, Def2, CARD_1:27;

then A5: the Element of Upper_Support (p,T,i) in Upper_Support (p,T,i) ;

then reconsider x9 = the Element of Upper_Support (p,T,i) 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