let n be Ordinal; :: thesis: for T being connected TermOrder of n
for p being Polynomial of n,L
for i being Element of NAT st i <= card () holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

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 () holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

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 () holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

let p be Polynomial of n,L; :: thesis: for i being Element of NAT st i <= card () holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

let i be Element of NAT ; :: thesis: ( i <= card () implies ( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) ) )

assume A1: i <= card () ; :: thesis: ( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

set l = Lower_Support (p,T,i);
thus Lower_Support (p,T,i) c= Support p by XBOOLE_1:36; :: thesis: ( card (Lower_Support (p,T,i)) = (card ()) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )

Upper_Support (p,T,i) c= Support p by ;
hence card (Lower_Support (p,T,i)) = (card ()) - (card (Upper_Support (p,T,i))) by CARD_2:44
.= (card ()) - i by ;
:: thesis: for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i)

now :: thesis: for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i)
let b, b9 be bag of n; :: thesis: ( b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T implies b9 in Lower_Support (p,T,i) )
assume that
A2: b in Lower_Support (p,T,i) and
A3: b9 in Support p and
A4: b9 <= b,T ; :: thesis: b9 in Lower_Support (p,T,i)
A5: b9 in (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) by A1, A3, Th19;
now :: thesis: b9 in Lower_Support (p,T,i)end;
hence b9 in Lower_Support (p,T,i) ; :: thesis: verum
end;
hence for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ; :: thesis: verum