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<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K2">--&gt;</a> 0;<br/>
<a NAME="E1:90_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K2">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K2">--&gt;</a> 0 as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E3:90_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:90_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:90_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:90_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:90_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>

<a NAME="E7:90_1_1"/>
 <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_1_1_1"/>
not  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:90_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:90_1_1_1"/>
 <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E5:90_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:90_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="polynom1.html#E3:90_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:90_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E3:90_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:90_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="polynom1.html#V1" target="_self">finite-support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D8" target="_self">Def8</a></i>;<br/>


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