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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">A</font>;<br/>



<b>set </b><font color="Maroon">cS</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E42</font></i>: 
 <a href="uproots.html#K4" target="_self">degree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E43</font></i>: 
 <a href="uproots.html#K4" target="_self">degree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:21"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E45</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:21_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T27">FINSEQ_2:27</a></i>;<br/><a NAME="E3:21_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E3:21_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:21_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:21_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E7:21_1_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><a NAME="E8:21_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E9:21_1_1"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E10:21_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E11:21_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 ;<br/><b>hence </b><a NAME="E12:21_1_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:21_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="rvsum_1.html#T115">RVSUM_1:115</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T20">BAGORDER:20</a></i>;<br/>


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