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<b>assume </b><a NAME="E1:68"/>
not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a> ,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="abcmiz_1.html#K20" title="ABCMIZ_1:func.20">Constructors</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:68"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a> ,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E4:68"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K20" title="ABCMIZ_1:func.20">Constructors</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:68"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="abcmiz_1.html#K17" title="ABCMIZ_1:func.17">Modes</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="abcmiz_1.html#K18" title="ABCMIZ_1:func.18">Attrs</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K19" title="ABCMIZ_1:func.19">Funcs</a>  )
 
<b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K17" title="ABCMIZ_1:func.17">Modes</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K18" title="ABCMIZ_1:func.18">Attrs</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K19" title="ABCMIZ_1:func.19">Funcs</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:68"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<b>consider </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:68"/><i><font color="Green" title="E42">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="txt" href="#E8:68"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:68"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a>  )
 
<b>by </b><i><a class="txt" href="#E3:68"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E12:68"/><b>then </b>
(  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<a NAME="E13:68"/><b>then </b>
(  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E14:68"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E15:68"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<b>hence </b><a NAME="E16:68"/>
contradiction
 <b>by </b><i><a class="txt" href="#E10:68"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="classes1.html#T33" title="CLASSES1:th.33">CLASSES1:33</a>, <a class="ref" href="classes1.html#T51" title="CLASSES1:th.51">CLASSES1:51</a></i>; <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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