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

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





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then </b><span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T18">ZFMISC_1:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1"/>
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then </b><span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="pboole.html#K3">/\</a> <span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">y</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E3:23_1_2_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T18">ZFMISC_1:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_2"/>
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
