let n be Ordinal; :: thesis: 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
for i being Element of NAT st i <= card (Support p) holds
for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T
let T be connected TermOrder of n; :: thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being Polynomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T
let p be Polynomial of n,L; :: thesis: for i being Element of NAT st i <= card (Support p) holds
for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T
let i be Element of NAT ; :: thesis: ( i <= card (Support p) implies for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T )
assume A1: i <= card (Support p) ; :: thesis: for b, b' being bag of st b in Upper_Support p,T,i & b' in Lower_Support p,T,i holds
b' < b,T
let b, b' be bag of ; :: thesis: ( b in Upper_Support p,T,i & b' in Lower_Support p,T,i implies b' < b,T )
assume A2: ( b in Upper_Support p,T,i & b' in Lower_Support p,T,i ) ; :: thesis: b' < b,T
A3: Lower_Support p,T,i c= Support p by XBOOLE_1:36;

now

assume b <= b',T ; :: thesis: contradiction
then b' in Upper_Support p,T,i by A1, A2, A3, Def2;
then b' in (Upper_Support p,T,i) /\ (Lower_Support p,T,i) by A2, XBOOLE_0:def 4;
hence contradiction by A1, Th19; :: thesis: verum

end;

hence b' < b,T by TERMORD:5; :: thesis: verum