let n be Ordinal; :: thesis:
let T be connected TermOrder of n; :: thesis:
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis:
let p be Polynomial of n,L; :: thesis:
let i be Element of NAT ; :: thesis:
assume A1: i <= card (Support p) ; :: thesis:
set l = Lower_Support p,T,i;
thus Lower_Support p,T,i c= Support p by XBOOLE_1:36; :: thesis:
Upper_Support p,T,i c= Support p by A1, Def2;
hence card (Lower_Support p,T,i) = (card (Support p)) - (card (Upper_Support p,T,i)) by CARD_2:63
.= (card (Support p)) - i by A1, Def2 ;
:: thesis:

let b, b' be bag of ; :: thesis:
assume A2: ( b in Lower_Support p,T,i & b' in Support p & b' <= b,T ) ; :: thesis:
then A3: b' in (Upper_Support p,T,i) \/ (Lower_Support p,T,i) by A1, Th19;

assume not b' in Lower_Support p,T,i ; :: thesis:
then b' in Upper_Support p,T,i by A3, XBOOLE_0:def 3;
then b < b',T by A1, A2, Th20;
hence contradiction by A2, TERMORD:5; :: thesis:

end;

hence b' in Lower_Support p,T,i ; :: thesis:

end;

hence for b, b' being bag of st b in Lower_Support p,T,i & b' in Support p & b' <= b,T holds
b' in Lower_Support p,T,i ; :: thesis: