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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</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/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="ordinal1.html#K5">omega</a> ;<br/>









<b>assume </b><a NAME="E1:36"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:36"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E5:36"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finset_1.html#T19">FINSET_1:19</a></i>;<br/>
<a NAME="E6:36"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T21">RELAT_1:21</a></i>;<br/>
<a NAME="E7:36"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_fund1.html#E5:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:36_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_fund1.html#E1:36_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="zf_fund1.html#E3:36_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:36_1"/><i><font color="Green">E48</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zf_fund1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:36_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_fund1.html#E4:36_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E7:36_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:36_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:36_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E9:36_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_fund1.html#E3:36_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="zf_fund1.html#E5:36_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E6:36_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_fund1.html#E7:36"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zf_fund1.html#T13" target="_self">Th13</a></i>;<br/>


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