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

<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>;<br/>
<b>set </b><font color="Maroon">XX</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>let </b><font color="Maroon">P</font> be   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">Q</font> be   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <font color="Olive">a</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Olive">c</font> )
 <b>and </b><br/><a NAME="E2:10_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff <font color="Olive">a</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Olive">c</font> )
 ;<br/>

<a NAME="E3:10_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1"><b>proof </b></a><div class="add">

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



<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_2_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:10_1_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font><b> such that </b><br/><a NAME="E4:10_1_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font><b> such that </b><br/><a NAME="E6:10_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E7:10_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <font color="Maroon">a</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:10_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_2_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:10_1_2_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font><b> such that </b><br/><a NAME="E4:10_1_2_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font><b> such that </b><br/><a NAME="E6:10_1_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E7:10_1_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff <font color="Maroon">a</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="tdgroup.html#NK1" target="_self">#</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:10_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


</div>
