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

<b>let </b><font color="Maroon">D</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">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:83"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">o</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>and </b><br/><a NAME="E2:83"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9">MONOID_0:def 9</a><br/>

<b>thus </b><a NAME="E3:83"/>
<font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a>  <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self">Def2</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9">MONOID_0:def 9</a><br/>

<b>set </b><font color="Maroon">a</font> =  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>;<br/>

<b>assume </b><a NAME="E4:83"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span>
 ;<br/>

<a NAME="E5:83"/><i><font color="Green">E46</font></i>: 
(  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp;  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self">Def2</a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:83"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T151">RELAT_1:151</a></i>;<br/>
<a NAME="E7:83"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:83"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E9:83"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font>, <font color="Maroon">a2</font> = <font color="Maroon">a2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:83"/>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E13:83"/><b>then </b>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:83"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a2</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:83"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E3:83_1"/>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E4:83_1"/><b>then </b>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:83_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:83_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:83"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E3:83_2"/>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E4:83_2"/><b>then </b>
<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:83_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:83_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:83"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">o</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E18:83"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self">Def2</a>, </i>;<br/>


</div>
