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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="polynom1.html#V1" target="_self">finite-support</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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="E1:139"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:139"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E4:139"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <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/>
<a NAME="E5:139"/>
 <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:139_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:139_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:139_1"/><b>then </b><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E4:139_1"/>
 <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E5:139_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:139"><i><font color="Green">E76</font></i></a>, <a class="txt" href="polynom1.html#E4:139"><i><font color="Green">E77</font></i></a>, <a class="txt" href="polynom1.html#E1:139_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="polynom1.html#E3:139_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E6:139_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:139"/><b>then </b>
 <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E7:139"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:139"><i><font color="Green">E76</font></i></a>, <a class="ref" href="polynom1.html#D8" target="_self">Def8</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:139"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self">Def14</a></i>;<br/>


</div>
