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

<b>let </b><font color="Maroon" title="c1">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> holds <br/> ( ( for <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font> ) iff <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font> ) iff <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<a NAME="E1:3"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d'</font> where <font color="Olive" title="b1">d'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : for <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="zf_colla.html#T1" target="_self" title="ZF_COLLA:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E2:3"/>
( ( for <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font> ) implies <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font> )</span><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" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:3_1"/><b>then </b>
for <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font> holds <br/> ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b2">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:3_1"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:3"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  holds not <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font></span><br/>

<b>given </b><font color="Maroon" title="c3">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><a NAME="E5:3"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:3"/>
 ex <font color="Olive" title="b1">d'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">d</font> &amp; ( for <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font> ) ) )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="zf_colla.html#E3:3"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<a NAME="E7:3"/><b>then </b>
 ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E5:3"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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