<?xml version="1.0"?>
<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">f</font> be   <a href="binop_1.html#NM2">BinOp</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>



<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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/>
<a NAME="E2:12"/><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>

<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E1:12_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:12_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:12"/><b>then </b>
for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
 
;<br/>
<a NAME="E7:12"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:12"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> st <font color="Olive">a</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a><br/>

<b>thus </b><a NAME="E9:12"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<b>thus </b><a NAME="E10:12"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>thus </b><a NAME="E11:12"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a><br/>

<b>thus </b><a NAME="E12:12"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>  ex <font color="Olive">r</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> st <br/>( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">l</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self">MONOID_0:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E1:12_3"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E13:12"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>  st ( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> or <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">a</font> ) holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D8" target="_self">MONOID_0:def 8</a><br/>

<b>thus </b><a NAME="E14:12"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> st <font color="Olive">a</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">f</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a>,<a class="ref" href="monoid_0.html#D9" target="_self">MONOID_0:def 9</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>

<b>thus </b><a NAME="E15:12"/>
( <font color="Maroon">f</font> <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> <font color="Maroon">f</font> implies ( <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> <font color="Maroon">f</font> ) )
 <b>by </b><i/>;<br/>


</div>
