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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">D</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">F</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><br/> for <font color="Olive" title="b2">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font><br/> for <font color="Olive" title="b3">I</font>, <font color="Olive" title="b4">J</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b3">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b4">J</font> holds <br/>( <font color="Olive" title="b2">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b3">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b2">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b4">J</font> &amp; <font color="Olive" title="b3">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b4">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">F</font> )</span><br/><b>let </b><font color="Maroon" title="c5">F</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"> for <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font><br/> for <font color="Olive" title="b2">I</font>, <font color="Olive" title="b3">J</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b2">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b3">J</font> holds <br/>( <font color="Olive" title="b1">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b2">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b1">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b3">J</font> &amp; <font color="Olive" title="b2">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b3">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> )</span><br/><b>let </b><font color="Maroon" title="c6">G</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>; <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">I</font>, <font color="Olive" title="b2">J</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b2">J</font> holds <br/>( <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b1">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Olive" title="b2">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> )</span><br/><b>let </b><font color="Maroon" title="c7">I</font>, <font color="Maroon" title="c8">J</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</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">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c8">J</font> implies ( <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c7">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> ) )</span><br/>













<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c7">I</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">J</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" 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">b<sub>1</sub></font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c8">J</font>  holds not <font color="Olive">b<sub>1</sub></font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> or ( <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c7">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> ) )</span><br/>

<b>given </b><font color="Maroon" title="c9">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c8">J</font><b> such that </b><a NAME="E3:52"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c9">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="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</font> &amp; <font color="Maroon" title="c7">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>thus </b><a NAME="E4:52"/>
<font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</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">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</font>
 <b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" 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">b<sub>1</sub></font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">J</font> st <font color="Olive">b<sub>1</sub></font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a></span><br/>

<b>take </b>
<font color="Maroon" title="c6">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">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="c6">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">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="c10">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D10" 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="c6">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a></span><br/>

<a NAME="E2:52_1"/>
<font color="Maroon" title="c7">I</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c8">J</font>
 
<b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E3:52_1"/><b>then </b><i><font color="Green" title="E38">A3</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="c7">I</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">J</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">b</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" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<i><font color="Green" title="E39">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font> = 
<font color="Maroon" title="c6">G</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:52_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E5:52_1"/>
<font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 
<b>by </b><i><a class="txt" href="#E3:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nattra_1.html#D10" title="NATTRA_1:def.10">NATTRA_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">b</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 <b>by </b><i><a class="txt" href="#E4:52_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="isocat_1.html#T3" target="_self" title="ISOCAT_1:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:52"/>
<font color="Maroon" title="c7">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" 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">b<sub>1</sub></font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">I</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">J</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> st <font color="Olive">b<sub>1</sub></font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a></span><br/>

<b>take </b>
<font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a></span><br/>

<b>let </b><font color="Maroon" title="c10">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" 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="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a></span><br/>

<a NAME="E6:52"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E7:52"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 <b>by </b><i><a class="txt" href="#E3:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nattra_1.html#D10" title="NATTRA_1:def.10">NATTRA_1:def 10</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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