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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>





<a NAME="E1:58"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E29</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> holds <br/><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp;  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E4:58_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:58_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E5:58_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><b>hence </b><a NAME="E6:58_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E1:58_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2"/><i><font color="Green">E44</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> holds <br/><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> or  not <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) )
 ;<br/><a NAME="E2:58_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:58_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E4:58_1_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:58_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_3"/><i><font color="Green">E48</font></i>: 
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp;  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 ;<br/><a NAME="E2:58_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:58_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E4:58_1_3"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:58_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_4"/><i><font color="Green">E50</font></i>: 
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; not <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> or  not <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) )
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:58_1_4"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E4:58_1_4"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="margrel1.html#T43">MARGREL1:43</a></i>;<br/><a NAME="E5:58_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E6:58_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>hence </b><a NAME="E7:58_1_4"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:58_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green">E52</font></i>: 
(  <a href="valuat_1.html#K5">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:58"/><i><font color="Green">E53</font></i>: 
(  <a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:58"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:58"/>
 <a href="valuat_1.html#K5">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_INF</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K22" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
