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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b> ex <font color="Olive">b1</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> ex <font color="Olive">Q1</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font> );<br/>
<a NAME="E1:14_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E2:14_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
;<br/>

<b>take </b>
<font color="Maroon">b1</font>
;<br/>

<b>take </b>
<font color="Maroon">b2</font>
;<br/>

<b>take </b>
<font color="Maroon">Q1</font>
;<br/>

<b>thus </b><a NAME="E4:14_1_1_1"/>
( <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="polynom1.html#NM3">Series</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font></span>)</span> where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> :  ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> ;<br/>
<a NAME="E5:14_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E6:14_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E7:14_1_1"/>
for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span>  holds <br/><font color="Olive">B</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E3:14_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font></span>)</span> where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> :  ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:14_1_1_2"/>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">Q0</font> = <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>) <b>-&gt; </b>   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> = <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">B0</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span> ;<br/>
<a NAME="E6:14_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_1_1_2_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E3:14_1_1_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E4:14_1_1_2_1_1"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:14_1_1_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E7:14_1_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:14_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:14_1_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E3:14_1_1_2_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E4:14_1_1_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font>
 ;<br/>
<b>thus </b><a NAME="E5:14_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:14_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E8:14_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b2</font>) where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> : <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q0</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<b>hence </b><a NAME="E9:14_1_1_2"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:14_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<a NAME="E9:14_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_1_1_3"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:14_1_1_3_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E4:14_1_1_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:14_1_1_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2'</font></span>)</span> where <font color="Olive">b2'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> :  ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Olive">b2'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> ) </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:14_1_1_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E9:14_1_1_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E10:14_1_1_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:14_1_1_3_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:14_1_1_3_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q1</font>
 
<b>by </b><i>, , , <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E13:14_1_1_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1_3_1_1"/>
<font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span>
 ;<br/><a NAME="E2:14_1_1_3_1_1"/><b>then </b>
<font color="Maroon">Q1</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:14_1_1_3_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E16:14_1_1_3_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:14_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:14_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14_1_1_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:14_1_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b1</font>) where <font color="Olive">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> : <font color="Olive">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E7:14_1_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font></span>)</span> where <font color="Olive">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> :  ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Olive">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E8:14_1_1_3"/>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E10:14_1_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E11:14_1_1_3"/><i><font color="Green">E83</font></i>: 
 ex <font color="Olive">Q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Olive">Q</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:14_1_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E14:14_1_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:14_1_1_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">Q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E17:14_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E18:14_1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E19:14_1_1_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E20:14_1_1_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">Q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E21:14_1_1_3"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E22:14_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E12:14_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">b1</font>
;<br/>

<b>take </b>
<font color="Maroon">b2</font>
;<br/>

<b>reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">Q1</font>
;<br/>

<b>thus </b><a NAME="E14:14_1_1"/>
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 ;<br/>

<b>thus </b><a NAME="E15:14_1_1"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>

<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E17:14_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">Q1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E18:14_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E19:14_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E20:14_1_1"/>
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2'</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E21:14_1_1"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 <b>by </b><i>, </i>;<br/>


</div>
