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<a NAME="E1:21_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="functor2.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="functor2.html#E2:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="functor2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E2:21_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="ref" href="functor2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:21_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor2.html#E2:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="functor2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K3" target="_self">`*`</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="functor2.html#E2:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="functor2.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7" target="_self">FUNCTOR2:def 7</a><br/>

<b>thus </b><a NAME="E2:21_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K3" target="_self">`*`</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K3" target="_self">`*`</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor2.html#K2" target="_self">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor2.html#E1:21_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="functor2.html#E2:21_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E3:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="functor2.html#E17"><i><font color="Green">Lemma15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K3" target="_self">`*`</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


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