<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b2">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b3">F3</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b3">F3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font>, <font color="Maroon" title="c5">F3</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c4">F2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> implies <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="nattra_1.html#D11" target="_self" title="NATTRA_1:def.11">NATTRA_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font>  holds not <font color="Olive" title="b1">t</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a>  or  not <font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> or <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font><span class="kw"> such that </span><a NAME="E3:54"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c6">t</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> or <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:54"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F3</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="nattra_1.html#D11" target="_self" title="NATTRA_1:def.11">NATTRA_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font>  holds not <font color="Olive" title="b1">t</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a>  or <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font><span class="kw"> such that </span><a NAME="E6:54"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c7">t9</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">F3</font></span><br/>

<span class="kw">thus </span><a NAME="E7:54"/>
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="nattra_1.html#D11" target="_self" title="NATTRA_1:def.11">NATTRA_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font> st <font color="Olive" title="b1">t</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c6">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c6">t</font> is  <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="nattra_1.html#D10" target="_self" title="NATTRA_1:def.10">NATTRA_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> </span><br/>

<a NAME="E8:54"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="nattra_1.html#D10" target="_self" title="NATTRA_1:def.10">Def10</a></span>;<br/>
<a NAME="E9:54"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="nattra_1.html#D10" target="_self" title="NATTRA_1:def.10">Def10</a></span>;<br/>
<a NAME="E10:54"/>
<font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="nattra_1.html#D7" target="_self" title="NATTRA_1:def.7">Def7</a></span>;<br/>
<a NAME="E11:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F3</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" target="_self" title="NATTRA_1:def.2">Def2</a></span>;<br/>
<a NAME="E12:54"/>
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="nattra_1.html#D7" target="_self" title="NATTRA_1:def.7">Def7</a></span>;<br/>
<a NAME="E13:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A8</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" target="_self" title="NATTRA_1:def.2">Def2</a></span>;<br/>
<a NAME="E14:54"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="nattra_1.html#T27" target="_self" title="NATTRA_1:th.27">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:54"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">a</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="cat_1.html#T75" title="CAT_1:th.75">CAT_1:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
