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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/>for <font color="Olive" title="b4">t</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font><br/> for <font color="Olive" title="b5">t'</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> holds  <font color="Olive" title="b5">t'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font> is    <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Olive" title="b1">F1</font>,<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 for <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font><br/> for <font color="Olive" title="b2">t'</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font> holds  <font color="Olive" title="b2">t'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b1">t</font> is    <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="nattra_1.html#R3" title="NATTRA_1:pred.3">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font> <a href="nattra_1.html#R3" title="NATTRA_1:pred.3">are_naturally_equivalent</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#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font><br/> for <font color="Olive" title="b2">t'</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font> holds  <font color="Olive" title="b2">t'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b1">t</font> is    <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font></span><br/>

<b>let </b><font color="Maroon" title="c6">t</font> be    <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</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">t'</font> being   <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</font> holds  <font color="Olive" title="b1">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#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font></span><br/><b>let </b><font color="Maroon" title="c7">t'</font> be    <a href="nattra_1.html#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">F3</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#M3" title="NATTRA_1:mode.3">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font></span><br/>



<b>thus </b><a NAME="E3:54"/>
<font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">F3</font> <a href="nattra_1.html#R3" title="NATTRA_1:pred.3">are_naturally_equivalent</a> 
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:54"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:54"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th32</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D14" target="_self" title="NATTRA_1:def.14">NATTRA_1:def 14</a> <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#V6" title="CAT_1:attr.6">invertible</a></span><br/>

<a NAME="E4:54"/><i><font color="Green" title="E46">A3</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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:54"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:54"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="nattra_1.html#D11" target="_self" title="NATTRA_1:def.11">Def11</a></i>;<br/>
<a NAME="E5:54"/><b>then </b><i><font color="Green" title="E47">A4</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#K9" title="CAT_1:func.9">*</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="ref" href="nattra_1.html#T27" target="_self" title="NATTRA_1:th.27">Th27</a></i>;<br/>
<a NAME="E6:54"/>
( <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> &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="nattra_1.html#E4:54"><i><font color="Green" title="E46">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="E7:54"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</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#K13" title="CAT_1:func.13">.</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#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</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#K13" title="CAT_1:func.13">.</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="E8:54"/>
( <font color="Maroon" title="c7">t'</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> &amp; <font color="Maroon" title="c6">t</font> is <a href="nattra_1.html#V1" title="NATTRA_1:attr.1">invertible</a> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:54"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:54"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="nattra_1.html#D14" target="_self" title="NATTRA_1:def.14">Def14</a></i>;<br/>
<a NAME="E9:54"/><b>then </b>
( <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#V6" title="CAT_1:attr.6">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#V6" title="CAT_1:attr.6">invertible</a> )
 
<b>by </b><i><a class="ref" href="nattra_1.html#D10" target="_self" title="NATTRA_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E10:54"/>
<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#V6" title="CAT_1:attr.6">invertible</a>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:54"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="nattra_1.html#E7:54"><i><font color="Green" title="E48">A5</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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