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<b>let </b><font color="Maroon">o1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:3"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff  ex <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">o</font> &amp; <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 ;<br/>

<a NAME="E2:3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:3_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_1_1"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">o</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:3_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o</font>
 <b>and </b><br/><a NAME="E4:3_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1"><b>suppose </b></a><a NAME="E1:3_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E3:3_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_2"><b>suppose </b></a><a NAME="E1:3_1_2_2"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:3_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_2_2_1_1"/>
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_2_2_1_1_1"/>
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_2_2_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">o</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="E3:3_1_2_2_1_1_2"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><a NAME="E4:3_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>;<br/><a NAME="E5:3_1_2_2_1_1_2"/>
<font color="Maroon">o</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font>
 <b>by </b><i>, , <a class="ref" href="ordinal3.html#T73">ORDINAL3:73</a></i>;<br/><a NAME="E6:3_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:3_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:3_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
