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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><font color="Maroon" title="c2">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">fs</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ;<br/>









<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">fs</font>,<font color="Maroon" title="c4">A</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fs</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:36"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E5:36"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finset_1.html#T19" title="FINSET_1:th.19">FINSET_1:19</a></i>;<br/>
<a NAME="E6:36"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="relat_1.html#T21" title="RELAT_1:th.21">RELAT_1:21</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" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:36_1"/><i><font color="Green" title="E45">A6</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E1:36_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="zf_fund1.html#E3:36_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E5:36_1"/><i><font color="Green" title="E46">A7</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T7" target="_self" title="ZF_FUND1:th.7">Th7</a></i>;<br/><a NAME="E6:36_1"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E4:36_1"><i><font color="Green" title="E45">A6</font></i></a></i>;<br/><a NAME="E7:36_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:36_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:36_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:36"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:36_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E3:36_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="zf_fund1.html#E5:36_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="zf_fund1.html#E6:36_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:36"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E6:36"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="zf_fund1.html#T13" target="_self" title="ZF_FUND1:th.13">Th13</a></i>;<br/>


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