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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><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="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="cat_2.html#K11">:]</a></span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>



<b>assume </b><a NAME="E3:48"/><i><font color="Green">E43</font></i>: 
(  <a href="isocat_2.html#K13" target="_self">Pr1</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K13" target="_self">Pr1</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="isocat_2.html#K14" target="_self">Pr2</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K14" target="_self">Pr2</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E4:48"/>
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="cat_2.html#D10">CAT_2:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="isocat_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self">Th35</a></i>
<br/>.= 
 <a href="isocat_2.html#K13" target="_self">Pr1</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="isocat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:48"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:48"><i><font color="Green">E42</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="cat_2.html#D10">CAT_2:def 10</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="cat_2.html#D11">CAT_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="isocat_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:48"><i><font color="Green">E41</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self">Th35</a></i>
<br/>.= 
 <a href="isocat_2.html#K14" target="_self">Pr2</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="isocat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:48"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:48"><i><font color="Green">E42</font></i></a>, <a class="ref" href="isocat_2.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="cat_2.html#D11">CAT_2:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E8:48"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E6:48"><i><font color="Green">E44</font></i></a>, <a class="ref" href="isocat_2.html#T5" target="_self">Th5</a></i>;<br/>


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