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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E43">A1</font></i>: 
<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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>given </b><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><b> such that </b><a NAME="E3:51"/><i><font color="Green" title="E44">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E4:51"/><i><font color="Green" title="E45">A3</font></i>: 
<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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>given </b><font color="Maroon" title="c7">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><b> such that </b><a NAME="E6:51"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c7">t'</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E7:51"/>
<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>
 <b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E4:51"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>take </b>
<font color="Maroon" title="c7">t'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">t'</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/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t'</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="E8:51"/><i><font color="Green" title="E47">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t'</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">t'</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>
 
<b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E4:51"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="nattra_1.html#T27" target="_self" title="NATTRA_1:th.27">Th27</a></i>;<br/>
<a NAME="E9:51"/>
( <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E4:51"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="nattra_1.html#D7" target="_self" title="NATTRA_1:def.7">Def7</a></i>;<br/>
<a NAME="E10:51"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
(  <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>  &amp;  <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>  )
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2" target="_self" title="NATTRA_1:def.2">Def2</a></i>;<br/>
<a NAME="E11:51"/>
( <font color="Maroon" title="c7">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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E3:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="#E6:51"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="nattra_1.html#D10" target="_self" title="NATTRA_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E12:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t'</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>
 <b>by </b><i><a class="txt" href="#E8:51"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="#E10:51"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="cat_1.html#T75" title="CAT_1:th.75">CAT_1:75</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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