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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>thus </b><a NAME="E1:66"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a><br/>

<b>thus </b><a NAME="E2:66"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T26" target="_self">Th26</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="catalg_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
 <a href="catalg_1.html#K9" target="_self">homsym</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>

<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T31" target="_self">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T27" target="_self">Th27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="catalg_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
 <a href="catalg_1.html#K9" target="_self">homsym</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>

<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E5:66"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>assume </b><a NAME="E7:66"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E8:66"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1"><b>suppose </b></a><a NAME="E1:66_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E4:66_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:66"><i><font color="Green">E35</font></i></a>, <a class="txt" href="catalg_1.html#E3:66_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="catalg_1.html#T28" target="_self">Th28</a>, <a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>

<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="catalg_1.html#T31" target="_self">Th31</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_2"><b>suppose </b></a><a NAME="E1:66_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:66_2_2"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:66"><i><font color="Green">E35</font></i></a>, <a class="txt" href="catalg_1.html#E3:66_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="catalg_1.html#T29" target="_self">Th29</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T145">FINSEQ_2:145</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="catalg_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="catalg_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalg_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>

<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:66_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="catalg_1.html#T32" target="_self">Th32</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
