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<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_2"/><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:33_1_2"/><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E3:33_1_2"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<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;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<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; <font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_3.html#R2">is_a_product_wrt</a>  <a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="cat_4.html#K8" target="_self">pr2</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="cat_4.html#T21" target="_self">Th21</a>, <a class="ref" href="cat_4.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:33_1_2"/><i><font color="Green">E23</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <span class="p1">(<span class="default"><a href="cat_4.html#K7" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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>6</sub></font> ) iff <font color="Maroon">c<sub>9</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_4.html#E1:33"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_3.html#T60">CAT_3:60</a></i>;<br/>
<a NAME="E6:33_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:33_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_4.html#E2:33_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_4.html#E5:33_1_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


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