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<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> 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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green">E43</font></i>: 
<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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green">E44</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> is <a href="cat_1.html#V6">terminal</a>
 ;<br/>

<a NAME="E3:76"/><i><font color="Green">E45</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</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>4</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#E1:76"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_3.html#D16" target="_self">Def16</a></i>;<br/>
<b>hence </b><a NAME="E4:76"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</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="cat_1.html#T19">CAT_1:19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17">CAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="cat_3.html#K10" target="_self">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E6:76"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_3.html#K10" target="_self">term</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_3.html#K10" target="_self">term</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:76"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E7:76"/><b>then </b>
(  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="cat_3.html#K10" target="_self">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</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="E9:76"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E10:76"/><i><font color="Green">E47</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> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_3.html#K10" target="_self">term</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) 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#E1:76"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cat_3.html#D16" target="_self">Def16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12">CAT_1:def 12</a><br/>

<b>thus </b><a NAME="E11:76"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cat_3.html#E9:76"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>

<b>thus </b><a NAME="E12:76"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E9:76"><i><font color="Green">E46</font></i></a>, <a class="txt" href="cat_3.html#E10:76"><i><font color="Green">E47</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><a NAME="E1:76_1"/><i><font color="Green">E49</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>10</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#E9:76"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E2:76_1"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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#E3:76"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E3:76_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <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>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cat_3.html#E12:76"><i><font color="Green">E48</font></i></a>, <a class="txt" href="cat_3.html#E1:76_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/><a NAME="E5:76_1"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cat_3.html#E2:76_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
 <a href="cat_3.html#K10" target="_self">term</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E2:76"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_3.html#T40" target="_self">Th40</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E2:76"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_3.html#T40" target="_self">Th40</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:76"/>
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:76"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cat_3.html#E3:76"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cat_3.html#T63" target="_self">Th63</a></i>;<br/>


</div>
