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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">B0</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> ;<br/>
<a NAME="E1:43"/><i><font color="Green">E42</font></i>: 
 <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:43_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:43_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T46">EQREL_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:43_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:43_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:43_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:43"/>
<span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</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="E1:43_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:43_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:43_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:43_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T46">EQREL_1:46</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
 <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
