<?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> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:71"/><i><font color="Green">E42</font></i>: 
(  <a href="cat_1.html#K6">Hom</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="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>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <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>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<b>thus </b><a NAME="E2:71"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</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="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</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>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) iff <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:71_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E2:71_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) iff <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:71_1"/><i><font color="Green">E44</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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>  )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E5:71_1"/><i><font color="Green">E45</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:71"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E6:71_1"/><b>then </b>
( <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>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E3:71_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E7:71_1"/><b>then </b><i><font color="Green">E46</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> ) iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:71_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:71_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>



<a NAME="E9:71_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E3:71_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cat_3.html#E5:71_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:71_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E12:71_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) iff <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E2:71_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_3.html#E11:71_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E14:71_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:71"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_3.html#E7:71_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E15:71_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) iff <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E12:71_1"><i><font color="Green">E48</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E3:71"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</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="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</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>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) iff <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) ) )
 ;<br/>

<b>thus </b><a NAME="E4:71"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:71"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E5:71"/><i><font color="Green">E44</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>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:71"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E7:71"/><i><font color="Green">E46</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:71"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E8:71"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</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="cat_3.html#E5:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cat_3.html#E6:71"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:71"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> ) iff <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:71"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<a NAME="E12:71"/><i><font color="Green">E48</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</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="cat_3.html#E3:71"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cat_3.html#E5:71"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cat_3.html#E6:71"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cat_3.html#E7:71"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:71"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E14:71"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E16:71"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:71"><i><font color="Green">E42</font></i></a>, <a class="txt" href="cat_3.html#E12:71"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E17:71"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> ) iff <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E10:71"><i><font color="Green">E47</font></i></a></i>;<br/>


</div>
