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<div class="add">

<a NAME="E1:32_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E2:32_1_1"/>
<font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>,<font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_1_1_1"/>
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:32_1_1_1"><i><font color="Green">E23</font></i></a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a><br/>

<a NAME="E2:32_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E3:32_1_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:32_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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#T126">CAT_1:126</a></i>;<br/>
<a NAME="E5:32_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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="txt" href="isocat_1.html#E4:32_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<a NAME="E6:32_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E7:32_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E8:32_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></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#T126">CAT_1:126</a></i>;<br/>
<a NAME="E9:32_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></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#T126">CAT_1:126</a></i>;<br/>
<a NAME="E10:32_1_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E11:32_1_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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="txt" href="isocat_1.html#E4:32_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<a NAME="E12:32_1_1_1"/><i><font color="Green">E85</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_1.html#E5:32_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E4:32_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nattra_1.html#T11">NATTRA_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_1.html#E1:32"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#T2" target="_self">Th2</a>, <a class="ref" href="isocat_1.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E3:32_1_1_1"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="nattra_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_1.html#E1:32"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E4:32_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nattra_1.html#T11">NATTRA_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#T4" target="_self">Th4</a>, <a class="ref" href="isocat_1.html#T7" target="_self">Th7</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_1_1"/>
<font color="Maroon">t</font> <a href="isocat_1.html#K3" target="_self">*</a> <font color="Maroon">F</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>,<font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 ;<br/>


</div>
