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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:43"/><i><font color="Green">E31</font></i>: 
 <a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V2" target="_self">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4" target="_self">ALTCAT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:43_1"/>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:43_1"/><b>then </b>
(  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E3:43_1"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>


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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:43"/>
 <a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V11" target="_self">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" target="_self">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D17" target="_self">ALTCAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:43_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:43_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:43_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E5:43_2"/><i><font color="Green">E35</font></i>: 
the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
;<br/>
<a NAME="E6:43_2"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E7:43_2"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E8:43_2"/><i><font color="Green">E38</font></i>: 
the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>
 
;<br/>
<a NAME="E9:43_2"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E10:43_2"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E3:43_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E11:43_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E5:43_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E12:43_2"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E8:43_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E13:43_2"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E8:43_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="altcat_1.html#E11:43_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E3:43_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E6:43_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_1.html#E7:43_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="altcat_1.html#E9:43_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E3:43_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E18:43_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E19:43_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="altcat_1.html#E11:43_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E20:43_2"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E21:43_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="altcat_1.html#E12:43_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<font color="Maroon">c<sub>20</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E11:43_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_1.html#E18:43_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:43_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_1.html#E11:43_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_1.html#E13:43_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="altcat_1.html#E19:43_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E12:43_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="altcat_1.html#E13:43_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="altcat_1.html#E21:43_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E12:43_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="altcat_1.html#E20:43_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
;<br/>


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

<b>thus </b><a NAME="E3:43"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="altcat_1.html#V5" target="_self">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" target="_self">ALTCAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" target="_self">ALTCAT_1:def 9</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E2:43_3"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>hence </b><a NAME="E3:43_3"/>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>assume </b><a NAME="E5:43_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:43_3"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:43_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:43_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E2:43_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E10:43_3"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E5:43_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:43_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E10:43_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" target="_self">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E5:43"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>hence </b><a NAME="E6:43"/>
 <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self">Th2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>assume </b><a NAME="E8:43"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1" target="_self">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E9:43"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E8:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E8:43"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E13:43"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E8:43"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" target="_self">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E13:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E13:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>


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