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<span class="kw">set </span><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/>
<span class="kw">thus </span><a NAME="E1:67_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="67_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E1:67_1_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_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="67_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:67_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>;<br/><a NAME="E3:67_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><span class="kw">thus </span>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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T15" target="_self" title="ALTCAT_1:th.15">Th15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E3:67_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="altcat_1.html#T16" target="_self" title="ALTCAT_1:th.16">Th16</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:67_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/><span class="kw">thus </span>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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="altcat_1.html#T28" target="_self" title="ALTCAT_1:th.28">Th28</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:67_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="67_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>

<a NAME="E1:67_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:67_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="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:67_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="67_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:67_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> 
  <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="67_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">j9</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/>
<span class="kw">take </span>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</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">j9</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">j9</font>,<font color="Maroon" title="c4">j9</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">j9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:67_1_1_3_1"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</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/>



<span class="kw">assume </span><a NAME="E4:67_1_1_3_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">i9</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="E6:67_1_1_3_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
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">i9</font>,<font color="Maroon" title="c4">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E7:67_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c7">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_3_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/>
<a NAME="E8:67_1_1_3_1"/><span class="kw">then </span>
<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">i9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_3_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E6:67_1_1_3_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>;<br/>
<a NAME="E9:67_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><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">j9</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">i9</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">i9</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">i9</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">j9</font></span>)</span>,<font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_1_1_3_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_1_1_3_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E9:67_1_1_3_1"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">j9</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/>
<span class="kw">take </span>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</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">j9</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">j9</font>,<font color="Maroon" title="c4">j9</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">j9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:67_1_1_3"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">j9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )</span><br/>



<span class="kw">assume </span><a NAME="E5:67_1_1_3"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">j9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">i9</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="E7:67_1_1_3"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
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">j9</font>,<font color="Maroon" title="c7">i9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E8:67_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c7">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_3"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/>
<a NAME="E9:67_1_1_3"/><span class="kw">then </span>
<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">i9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_3"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E7:67_1_1_3"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>;<br/>
<a NAME="E10:67_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><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">j9</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">i9</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">i9</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">i9</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">j9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_1_1_3"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_1_1_3"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="txt" href="#E10:67_1_1_3"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:67_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c6">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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">i</font>,<font color="Maroon" title="c4">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="c4">j</font>,<font color="Maroon" title="c5">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="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">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="c4">j</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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/>

<span class="kw">set </span><font color="Maroon" title="c7">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/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">i9</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c10">j9</font> = <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c11">k9</font> = <font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c12">l9</font> = <font color="Maroon" title="c6">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/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">i</font>,<font color="Maroon" title="c4">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="c4">j</font>,<font color="Maroon" title="c5">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="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">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="c4">j</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:67_1_1_4"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<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="c3">i</font>,<font color="Maroon" title="c4">j</font>
 <span class="kw">and </span><br/><a NAME="E3:67_1_1_4"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<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="c4">j</font>,<font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E4:67_1_1_4"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<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="c5">k</font>,<font color="Maroon" title="c6">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">i</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">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="c4">j</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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="E5:67_1_1_4"/>
<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">i9</font>,<font color="Maroon" title="c10">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1_4"><span class="lab"><font color="Green" title="E39">A15</font></span></a></span>;<br/>
<a NAME="E6:67_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<font color="Maroon" title="c9">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/>
<a NAME="E7:67_1_1_4"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">i9</font>,<font color="Maroon" title="c9">i9</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">i9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D22" target="_self" title="ALTCAT_1:def.22">Def22</a></span>;<br/>
<a NAME="E8:67_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1_4"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E6:67_1_1_4"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:67_1_1_4"/>
<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">j9</font>,<font color="Maroon" title="c11">k9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_4"><span class="lab"><font color="Green" title="E40">A16</font></span></a></span>;<br/>
<a NAME="E10:67_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c10">j9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/>
<a NAME="E11:67_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_4"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E6:67_1_1_4"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E7:67_1_1_4"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E12:67_1_1_4"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
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">i9</font>,<font color="Maroon" title="c9">i9</font>,<font color="Maroon" title="c9">i9</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">i9</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">i9</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">i9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T29" target="_self" title="ALTCAT_1:th.29">Th29</a></span>;<br/>
<a NAME="E13:67_1_1_4"/>
<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">k9</font>,<font color="Maroon" title="c12">l9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_4"><span class="lab"><font color="Green" title="E41">A17</font></span></a></span>;<br/>
<a NAME="E14:67_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c11">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">l9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/>
<a NAME="E15:67_1_1_4"/><span class="kw">then </span>
<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">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_4"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E6:67_1_1_4"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E10:67_1_1_4"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E7:67_1_1_4"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:67_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="c3">i</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c5">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="c3">i</font>,<font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c6">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="c4">j</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_4"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E10:67_1_1_4"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E14:67_1_1_4"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:67_1_1_4"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E11:67_1_1_4"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E12:67_1_1_4"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
