
theorem Th24:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non empty addLoopStr, p
  being Polynomial of n,L, i being Element of NAT st i <= card(Support p) holds
Lower_Support(p,T,i) c= Support p & card Lower_Support(p,T,i) = card(Support p)
- 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)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non empty addLoopStr, p be Polynomial of n,
  L, i being Element of NAT;
  assume
A1: i <= card Support p;
  set l = Lower_Support(p,T,i);
  thus l c= Support p by XBOOLE_1:36;
  Upper_Support(p,T,i) c= Support p by A1,Def2;
  hence card l = card(Support p)-card(Upper_Support(p,T,i)) by CARD_2:44
    .= card(Support p) - i by A1,Def2;
  now
    let b,b9 be bag of n;
    assume that
A2: b in Lower_Support(p,T,i) and
A3: b9 in Support p and
A4: b9 <= b,T;
A5: b9 in Upper_Support(p,T,i) \/ Lower_Support(p,T,i) by A1,A3,Th19;
    now
      assume not b9 in Lower_Support(p,T,i);
      then b9 in Upper_Support(p,T,i) by A5,XBOOLE_0:def 3;
      then b < b9,T by A1,A2,Th20;
      hence contradiction by A4,TERMORD:5;
    end;
    hence b9 in Lower_Support(p,T,i);
  end;
  hence thesis;
end;
