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

<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> , <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">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>let </b><font color="Maroon">X</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>;<br/>











<b>assume </b><a NAME="E1:33"/><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; ( <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) )
 ;<br/>

<b>assume </b><a NAME="E2:33"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> is <a href="binop_1.html#V1">commutative</a>
 ;<br/>

<a NAME="E3:33"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:33"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>) ) )
 <b>from </b><i><a class="ref" href="funct_3.html#S2">FUNCT_3:sch 2</a>();<br/></i>
<a NAME="E6:33"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z2</font> )
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E6:33_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:33_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:33_1_1"/>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:33_1_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:33_1_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1"/>
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<a NAME="E1:33_2"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:33_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_2_1_1"/><i><font color="Green">E76</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">y1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E4:33_2_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_2_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:33_2_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:33_2_1_1"/>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E8:33_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>assume </b><a NAME="E2:33_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:33_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:33_2_1"/><i><font color="Green">E81</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">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:33_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:33_2_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:33_2_1"/><b>then </b>
<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">y1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:33_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_2"/>
<font color="Maroon">h</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:33"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><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>assume </b><a NAME="E1:33_3"/><i><font color="Green">E85</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"><font color="Maroon">J</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_3"/><i><font color="Green">E86</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">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E4:33_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_3"/><b>then </b>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:33_3"/><b>then </b>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <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>hence </b><a NAME="E7:33_3"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</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="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>proof </b></a><div class="add">

<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="E2:33_4"/><i><font color="Green">E89</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/>
<a NAME="E3:33_4"/><i><font color="Green">E90</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">g</font>,<font color="Maroon">f</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">J</font>,<font color="Maroon">I</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="E4:33_4"/>
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> ) ) ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_4_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_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>thus </b><a NAME="E1:33_4_1_1"/>
( <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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_4_1_1_1"/><i><font color="Green">E91</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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_4_1_1_1"/><i><font color="Green">E92</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">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E4:33_4_1_1_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_4_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:33_4_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <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="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#T106">ZFMISC_1:106</a>, </i>;<br/>


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

<b>assume </b><a NAME="E2:33_4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E3:33_4_1_1"/>
<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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:33_4_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_2"><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>assume </b><a NAME="E1:33_4_1_2"/><i><font color="Green">E119</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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_4_1_2"/><i><font color="Green">E120</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">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:33_4_1_2"/><i><font color="Green">E121</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <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="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><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">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><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="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">y</font>
<b>by </b><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">z</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/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <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>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</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">g</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="finseqop.html#T82">FINSEQOP:82</a></i>
;<br/>


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

<b>thus </b><a NAME="E3:33_4_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_4"/>
<font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</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="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</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">J</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<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#T5">FUNCT_2:5</a></i>;<br/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<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="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="setwop_2.html#T8">SETWOP_2:8</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_3.html#K9" target="_self">*</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E12:33"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></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="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></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">g</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>


</div>
