<?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">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">J</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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>, <font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">J</font>,<font color="Maroon">D</font>;<br/>









<b>assume </b><a NAME="E1:34"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font> ;<br/>
<a NAME="E3:34"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </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">I</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/><a NAME="E1:34_1"/><i><font color="Green">E45</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</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>reconsider </b><font color="Maroon">u</font> = <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> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</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/><i><font color="Green">E48</font></i>: <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">u</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <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>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseqop.html#T82">FINSEQOP:82</a></i>
;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/><a NAME="E5:34_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>

;<br/><a NAME="E7:34_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:34_1"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E9:34_1"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E10:34_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E11:34_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:34_1"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:34_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T71">FUNCT_3:71</a></i>;<br/><a NAME="E14:34_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:34_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:34_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">z</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E17:34_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#D4">FUNCOP_1:def 4</a></i>;<br/><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T35">FUNCOP_1:35</a></i>
;<br/><b>hence </b><a NAME="E19:34_1"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><br/> for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">y</font></span><a href="setwiseo.html#K2">.}</a></span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">y</font></span><a href="setwiseo.html#K2">.}</a></span>,<font color="Maroon">g</font></span>)</span></span>)</span>
 ;<br/>


</div>
