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 Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low p,T,i) c= Support (Red p,T)
let T be connected TermOrder of n; :: thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low p,T,i) c= Support (Red p,T)
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low p,T,i) c= Support (Red p,T)
let p be Polynomial of n,L; :: thesis: for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low p,T,i) c= Support (Red p,T)
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= card (Support p) implies Support (Low p,T,i) c= Support (Red p,T) )
assume that
A1: 1 <= i and
A2: i <= card (Support p) ; :: thesis: Support (Low p,T,i) c= Support (Red p,T)
Support p <> {} by A1, A2;
then p <> 0_ n,L by POLYNOM7:1;
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def 2;
set sl = Lower_Support p,T,i;

A3: now

assume A4: HT p,T in Lower_Support p,T,i ; :: thesis: contradiction
HT p,T in Upper_Support p,T,i by A1, A2, Th23;
then HT p,T in (Upper_Support p,T,i) /\ (Lower_Support p,T,i) by A4, XBOOLE_0:def 4;
hence contradiction by A2, Th19; :: thesis: verum

end;

now

consider u being Element of {(HT p,T)} /\ (Lower_Support p,T,i);
assume {(HT p,T)} /\ (Lower_Support p,T,i) <> {} ; :: thesis: contradiction
then ( u in {(HT p,T)} & u in Lower_Support p,T,i ) by XBOOLE_0:def 4;
hence contradiction by A3, TARSKI:def 1; :: thesis: verum

end;

then {(HT p,T)} misses Lower_Support p,T,i by XBOOLE_0:def 7;
then A5: (Lower_Support p,T,i) \ {(HT p,T)} = Lower_Support p,T,i by XBOOLE_1:83
.= Support (Low p,T,i) by A2, Lm3 ;
(Support (Low p,T,i)) \ {(HT p,T)} c= (Support p) \ {(HT p,T)} by A2, Th26, XBOOLE_1:33;
then (Support (Low p,T,i)) \ {(HT p,T)} c= Support (Red p,T) by TERMORD:36;
hence Support (Low p,T,i) c= Support (Red p,T) by A2, A5, Lm3; :: thesis: verum