<?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> , <font color="Maroon">b</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/>



<a NAME="E1:32"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <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>   st <font color="Olive">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Olive">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">a</font> <a href="funct_2.html#R2">=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<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> , <font color="Maroon">b</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>assume </b><a NAME="E1:32_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:32_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> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:32_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_1_1"/><b>then </b>
 <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:32"><i><font color="Green">E19</font></i></a>, </i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<b>hence </b><a NAME="E4:32_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds <br/> ( <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:32_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:32_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E2:32_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1_2"/>
( <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_1"/><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> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3"><b>proof </b></a><div class="add">

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

<a NAME="E1:32_1_3"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_1_3"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:32_1_3"/>
( (  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or (  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or (  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a>, <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_3"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><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="E6:32_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <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="E8:32_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b</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="E9:32_1"/>
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="E10:32_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <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>   st <font color="Olive">a</font> <a href="funct_2.html#R2">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Olive">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<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> , <font color="Maroon">b</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>assume </b><a NAME="E1:32_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:32_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds <br/> ( <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:32_2_1"/>
( <font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  iff  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
;<br/>
<a NAME="E2:32_2_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2_1"/>
( <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:32_2_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_2_2"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:32_2_2"/>
 <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:32_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><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="E5:32_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font> <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="E7:32_2"/><i><font color="Green">E76</font></i>: 
(  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</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="E8:32_2"/>
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="E9:32_2"/>
<font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32"/>
( <font color="Maroon">a</font> <a href="bvfunc_1.html#K13" target="_self">'eqv'</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> iff <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
