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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:59"/>
<span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<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="E1:59_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:59_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T29" target="_self">Th29</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<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#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:59_1"/>
not <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="monoid_1.html#E2:59_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:59_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_1.html#K15" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:59_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="monoid_1.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E7:59_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></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="E2:59"/><b>then </b>
<span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> 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/>
<b>hence </b><a NAME="E3:59"/>
 <a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K19" target="_self">finite-MultiSet_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="monoid_1.html#D6" target="_self">Def6</a></i>;<br/>


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