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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">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="c3">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c2">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) )
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c3">B</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#E1:35"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E43">A5</font></i>: 
( <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:35"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E3:35_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:35"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:35"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E1:35_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="zf_fund1.html#T4" target="_self" title="ZF_FUND1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:35"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">B</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="zf_fund1.html#E3:35"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="zf_fund1.html#E2:35"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E4:35"><i><font color="Green" title="E42">A4</font></i></a>);</i><br/>


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