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<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>



<a NAME="E1:3"/><i><font color="Green">E33</font></i>: 
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">d'</font> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Olive">d'</font> holds <br/> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:3"/>
( ( for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  holds not <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> ) implies <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/>
for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  holds not <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> holds <br/> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:3_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>given </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><a NAME="E4:3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E5:3"/>
 ex <font color="Olive">d'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Olive">d'</font> holds <br/> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:3"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:3"/>
contradiction
 ;<br/>


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