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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>







<b>assume </b><a NAME="E1:38"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x2</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1"/>
( <font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x2</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> or <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>suppose </b></a><a NAME="E1:38_1_1"/>
<font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font>
 ;<br/><div class="add"><a NAME="E2:38_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pboole.html#T5">PBOOLE:5</a></i>;<br/><a NAME="E3:38_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><b>suppose </b></a><a NAME="E1:38_1_2"/>
<font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><a NAME="E2:38_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</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/><a NAME="E3:38_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><b>hence </b><a NAME="E4:38_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_3"><b>suppose </b></a><a NAME="E1:38_1_3"/>
<font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x2</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>
 ;<br/><div class="add"><a NAME="E2:38_1_3"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</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/><a NAME="E3:38_1_3"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><b>hence </b><a NAME="E4:38_1_3"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_4"><b>suppose </b></a><a NAME="E1:38_1_4"/>
<font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_4"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pzfmisc1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="pzfmisc1.html#K2" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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