<?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>thus </b><a NAME="E1:8"/>
( <font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="mboolean.html#K1" target="_self">bool</a> <font color="Maroon">Y</font> implies for <font color="Olive">A</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Olive">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font> iff <font color="Olive">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="mboolean.html#K1" target="_self">bool</a> <font color="Maroon">Y</font>
 ;<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>thus </b><a NAME="E2:8_1"/>
( <font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font> implies <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font>
 ;<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="E2:8_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E3:8_1_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, , , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:8_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> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


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

<b>assume </b><a NAME="E3:8_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 ;<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#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E4:8_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E5:8_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</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="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:8"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">A</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Olive">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font> iff <font color="Olive">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><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="E1:8_2"/><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/><a NAME="E2:8_2"/>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T49">PBOOLE:49</a></i>;<br/><a NAME="E3:8_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E4:8_2"/>
for <font color="Olive">A'</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> iff <font color="Olive">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:8_2_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E2:8_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:8_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E38</font></i>: <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">A'</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<b>thus </b><a NAME="E6:8_2_1"/>
( <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> implies <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:8_2_1_2"/>
<font color="Maroon">K</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:8_2_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1_2_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:8_2_1_2_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_2_1_2_1_1_1_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:8_2_1_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:8_2_1_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:8_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:8_2_1_2_1_1_1_2"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_2_1_2_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:8_2_1_2"/><b>then </b>
<font color="Maroon">K</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:8_2_1_2"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</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="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E7:8_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E8:8_2_1"/>
<font color="Maroon">K</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</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="E1:8_2_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1_3_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_3_1_1_1"><b>case </b></a><a NAME="E1:8_2_1_3_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_2_1_3_1_1_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_3_1_1_2"><b>case </b></a><a NAME="E1:8_2_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:8_2_1_3_1_1_2"/><b>then </b>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">A'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:8_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E4:8_2_1_3_1_1_2"/>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T49">PBOOLE:49</a></i>;<br/><b>hence </b><a NAME="E5:8_2_1_3_1_1_2"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_2_1_3"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:8_2_1"/><b>then </b>
<font color="Maroon">K</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:8_2_1"/>
<font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:8_2"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:8_2"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mboolean.html#K1" target="_self">bool</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="mboolean.html#K1" target="_self">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
