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<b>consider </b><font color="Maroon">D</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">W</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">W</font> = <font color="Maroon">D</font>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font> &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">L1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S4">ORDINAL1:sch 4</a>();<br/></i>
<a NAME="E4:27_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:27_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:27_1_1_1"/>
<font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E6:27_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:27_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="ordinal1.html#M1">T-Sequence</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E6:27_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D5" target="_self">ZF_REFLE:def 5</a><br/>

<b>assume </b><a NAME="E7:27_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a><br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:27_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E11:27_1_1"/>
<font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E12:27_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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