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

<b>set </b><font color="Maroon">C</font> =  <a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font>;<br/>
<b>thus </b><a NAME="E1:65_1_1"/>
 <a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font> is <a href="altcat_1.html#V10" target="_self">pseudo-functional</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D15" target="_self">ALTCAT_1:def 15</a><br/>

<a NAME="E1:65_1_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o1</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_1_1"/>
( ( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o3</font> ) or <font color="Maroon">o1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o2</font> or <font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o3</font> )
 ;<br/><div class="add"><a NAME="E2:65_1_1_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/><a NAME="E3:65_1_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font> )
 <b>by </b><i>, <a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>thus </b>the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span> <a href="funct_2.html#K4">:-&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, </i>
<br/>.= 
 <a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">o1</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="altcat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="altcat_1.html#T4" target="_self">Th4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o2</font> or <font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o3</font> )
 ;<br/><div class="add"><a NAME="E2:65_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:65_1_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><b>thus </b>the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="altcat_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:65_1_1"/>
 <a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font> is <a href="altcat_1.html#V14" target="_self">pseudo-discrete</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D21" target="_self">ALTCAT_1:def 21</a><br/>

<a NAME="E1:65_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">i</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:65_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">i</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a>
 ;<br/>


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

<b>thus </b><a NAME="E3:65_1_1"/>
 <a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font> is <a href="altcat_1.html#V12" target="_self">with_units</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:65_1_1_3"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> is <a href="altcat_1.html#V5" target="_self">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" target="_self">ALTCAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" target="_self">ALTCAT_1:def 9</a><br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font>
;<br/>

the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>

<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:65_1_1_3_1"/>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:65_1_1_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<a NAME="E6:65_1_1_3_1"/><i><font color="Green">E91</font></i>: 
the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
;<br/>
<a NAME="E7:65_1_1_3_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:65_1_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E9:65_1_1_3_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_2.html#K4">:-&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="altcat_1.html#T11" target="_self">Th11</a>, </i>
;<br/>


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

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" target="_self">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font>
;<br/>

the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>

<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:65_1_1_3"/>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E5:65_1_1_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<a NAME="E7:65_1_1_3"/><i><font color="Green">E102</font></i>: 
the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">i'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
;<br/>
<a NAME="E8:65_1_1_3"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:65_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E10:65_1_1_3"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_2.html#K4">:-&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="altcat_1.html#T11" target="_self">Th11</a>, </i>
;<br/>


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

<b>thus </b><a NAME="E4:65_1_1"/>
 <a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font> is <a href="altcat_1.html#V11" target="_self">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">G</font> = the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</font> = the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">l</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" target="_self">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D17" target="_self">ALTCAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font>, <font color="Maroon">l'</font> = <font color="Maroon">l</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:65_1_1_4"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">l</font> )
 ;<br/>

<a NAME="E3:65_1_1_4"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">k'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">k'</font>,<font color="Maroon">l'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> )
 
;<br/>
<a NAME="E4:65_1_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:65_1_1_4"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">i'</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:65_1_1_4"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font> )
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:65_1_1_4"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i'</font>,<font color="Maroon">i'</font>,<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_2.html#K4">:-&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:65_1_1_4"/><b>then </b>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i'</font>,<font color="Maroon">i'</font>,<font color="Maroon">i'</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <font color="Maroon">i'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:65_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" target="_self">DiscrCat</a> <font color="Maroon">A</font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="altcat_1.html#T10" target="_self">Th10</a></i>;<br/>


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


</div>
