<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1
;<br/>

<a NAME="E2:11_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:11_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:11_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E7:11_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:11_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:11_1_1"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:11_1_1_3_1"/><b>then </b><i><font color="Green">E38</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:11_1_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E5:11_1_1_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:11_1_1"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,1 is <a href="polynom7.html#V2" target="_self">univariate</a>
 <b>by </b><i/>;<br/>


</div>
