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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><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>, <font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7">ALTCAT_1:def 7</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:26"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>assume </b><a NAME="E4:26"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:26"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>assume </b><a NAME="E7:26"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</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>
 ;<br/>

<a NAME="E8:26"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</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_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E15</font></i>: 
 <a href="cat_1.html#K6">Hom</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="txt" href="altcat_2.html#E2:26"><i><font color="Green">E12</font></i></a>, <a class="txt" href="altcat_2.html#E5:26"><i><font color="Green">E13</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>
<a NAME="E11:26"/><i><font color="Green">E16</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</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="txt" href="altcat_2.html#E5:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="altcat_2.html#E8:26"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>
<a NAME="E12:26"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E2:26"><i><font color="Green">E12</font></i></a>, <a class="txt" href="altcat_2.html#E5:26"><i><font color="Green">E13</font></i></a>, <a class="ref" href="altcat_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E13:26"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="altcat_2.html#E8:26"><i><font color="Green">E14</font></i></a>, <a class="ref" href="altcat_2.html#T14" target="_self">Th14</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E8:26"><i><font color="Green">E14</font></i></a>, <a class="txt" href="altcat_2.html#E10:26"><i><font color="Green">E15</font></i></a>, <a class="txt" href="altcat_2.html#E12:26"><i><font color="Green">E17</font></i></a>, <a class="ref" href="altcat_2.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E2:26"><i><font color="Green">E12</font></i></a>, <a class="txt" href="altcat_2.html#E5:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="altcat_2.html#E8:26"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K5" target="_self">the_comps_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="altcat_2.html#E2:26"><i><font color="Green">E12</font></i></a>, <a class="txt" href="altcat_2.html#E11:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="altcat_2.html#E13:26"><i><font color="Green">E18</font></i></a>, <a class="ref" href="altcat_2.html#T14" target="_self">Th14</a></i>
;<br/>


</div>
