
theorem Th27:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
being Polynomial of n,L, i being Element of NAT st 1 <= i & i <= card(Support p
  ) holds Support(Low(p,T,i)) c= Support(Red(p,T))
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_complementable right_zeroed non trivial addLoopStr, p be Polynomial of
  n,L, i be Element of NAT;
  assume that
A1: 1 <= i and
A2: i <= card(Support p);
  Support p <> {} by A1,A2;
  then p <> 0_(n,L) by POLYNOM7:1;
  then reconsider p as non-zero Polynomial of n,L by POLYNOM7:def 1;
  set sl = Lower_Support(p,T,i);
A3: now
    assume
A4: HT(p,T) in sl;
    HT(p,T) in Upper_Support(p,T,i) by A1,A2,Th23;
    then HT(p,T) in Upper_Support(p,T,i) /\ sl by A4,XBOOLE_0:def 4;
    hence contradiction by A2,Th19;
  end;
  now
    set u = the Element of {HT(p,T)} /\ sl;
    assume {HT(p,T)} /\ sl <> {};
    then u in {HT(p,T)} & u in sl by XBOOLE_0:def 4;
    hence contradiction by A3,TARSKI:def 1;
  end;
  then {HT(p,T)} misses sl by XBOOLE_0:def 7;
  then
A5: sl \ {HT(p,T)} = sl 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 thesis by A2,A5,Lm3;
end;
