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<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>thus </b><a NAME="E1:64_1_1"/>
 <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V10" title="ALTCAT_1:attr.10">pseudo-functional</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V14" title="ALTCAT_1:attr.14">pseudo-discrete</a> &amp;  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> &amp;  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c5">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D15" target="_self" title="ALTCAT_1:def.15">ALTCAT_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E1:64_1_1_1"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c3">o1</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_1_1_1_1"/>
( ( <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o3</font> ) or <font color="Maroon" title="c3">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">o2</font> or <font color="Maroon" title="c4">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">o3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_1"/><i><font color="Green" title="E40">A2</font></i>: 
( <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>;<br/><a NAME="E3:64_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> )
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:64_1_1_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="altcat_1.html#E1:64_1_1_1_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>thus </b>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E1:64_1_1_1_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></i>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#T15" target="_self" title="ALTCAT_1:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:64_1_1_1_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="altcat_1.html#E3:64_1_1_1_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="altcat_1.html#T16" target="_self" title="ALTCAT_1:th.16">Th16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_2"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c3">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">o2</font> or <font color="Maroon" title="c4">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">o3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></i>;<br/><a NAME="E3:64_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>thus </b>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:64_1_1_1_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="altcat_1.html#T28" target="_self" title="ALTCAT_1:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:64_1_1_1_1_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:64_1_1"/>
 <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V14" title="ALTCAT_1:attr.14">pseudo-discrete</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> &amp;  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D21" target="_self" title="ALTCAT_1:def.21">ALTCAT_1:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c3">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a></span><br/>

<a NAME="E1:64_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c3">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c3">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>;<br/>
<b>hence </b><a NAME="E2:64_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c3">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:64_1_1"/>
 <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:64_1_1_3"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" target="_self" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" target="_self" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c3">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">j'</font> = <font color="Maroon" title="c3">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">j'</font>,<font color="Maroon" title="c4">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>

<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>
;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_1"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )</span><br/>



<b>assume </b><a NAME="E4:64_1_1_3_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c5">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E6:64_1_1_3_1"/><i><font color="Green" title="E40">A8</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i'</font>,<font color="Maroon" title="c4">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E7:64_1_1_3_1"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c7">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j'</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:64_1_1_3_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></i>;<br/>
<a NAME="E8:64_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:64_1_1_3_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="altcat_1.html#E6:64_1_1_3_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>;<br/>
<a NAME="E9:64_1_1_3_1"/><b>then </b><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Maroon" title="c6">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>,<font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E7:64_1_1_3_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E7:64_1_1_3_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="altcat_1.html#E9:64_1_1_3_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c3">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" target="_self" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">j'</font> = <font color="Maroon" title="c3">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">j'</font>,<font color="Maroon" title="c4">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>

<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>
;<br/>
<b>hence </b><a NAME="E4:64_1_1_3"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )</span><br/>



<b>assume </b><a NAME="E5:64_1_1_3"/><i><font color="Green" title="E39">A11</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c5">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E7:64_1_1_3"/><i><font color="Green" title="E40">A12</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">j'</font>,<font color="Maroon" title="c7">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E8:64_1_1_3"/><b>then </b><i><font color="Green" title="E41">A13</font></i>: 
<font color="Maroon" title="c7">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j'</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E5:64_1_1_3"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></i>;<br/>
<a NAME="E9:64_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E5:64_1_1_3"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="altcat_1.html#E7:64_1_1_3"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>;<br/>
<a NAME="E10:64_1_1_3"/><b>then </b><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">i'</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E8:64_1_1_3"><i><font color="Green" title="E41">A13</font></i></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E8:64_1_1_3"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="altcat_1.html#E10:64_1_1_3"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E4:64_1_1"/>
 <a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">c</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c8">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" target="_self" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D17" target="_self" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c11">k'</font> = <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c12">l'</font> = <font color="Maroon" title="c8">l</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon" title="c13">f</font>, <font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c15">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font> )</span><br/>

<b>assume </b><a NAME="E2:64_1_1_4"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c15">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font></span><br/>

<a NAME="E3:64_1_1_4"/><b>then </b>
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c10">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">j'</font>,<font color="Maroon" title="c11">k'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c15">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">k'</font>,<font color="Maroon" title="c12">l'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
;<br/>
<a NAME="E4:64_1_1_4"/><b>then </b><i><font color="Green" title="E40">A16</font></i>: 
( <font color="Maroon" title="c9">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j'</font> &amp; <font color="Maroon" title="c10">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k'</font> &amp; <font color="Maroon" title="c11">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">l'</font> )
 
<b>by </b><i><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></i>;<br/>
<a NAME="E5:64_1_1_4"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c9">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></i>;<br/>
<a NAME="E6:64_1_1_4"/><b>then </b><i><font color="Green" title="E41">A17</font></i>: 
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font> &amp; <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font> )
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:64_1_1_4"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="altcat_1.html#E4:64_1_1_4"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:64_1_1_4"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c9">i'</font>,<font color="Maroon" title="c9">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font></span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E8:64_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K9" title="ALTCAT_1:func.9">DiscrCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c8">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E4:64_1_1_4"><i><font color="Green" title="E40">A16</font></i></a>, <a class="txt" href="altcat_1.html#E6:64_1_1_4"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
