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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">l</font> =  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>;<br/>
<b>thus </b><a NAME="E2:31"/>
 <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:31"/>
 <a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b'</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b'</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 ;<br/><a NAME="E2:31_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1"/>
not <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 ;<br/><a NAME="E2:31_2_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:31_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="termord.html#R2">&lt;</a> <font color="Maroon">b'</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:31_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
for <font color="Olive">b</font>, <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">b'</font> <a href="termord.html#R1">&lt;=</a> <font color="Olive">b</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">i</font>
 ;<br/>


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