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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</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="firebrick">::  thesis: </font></span></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="c3">B</font><br/>  for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font><br/>  for <font color="Olive" title="b4">H</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b5">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">G1</font>,<font color="Olive" title="b3">G2</font>  st <font color="Olive" title="b2">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">G2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b5">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Olive" title="b1">F</font></span>)</span></span><br/><span class="kw">let </span><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="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font><br/>  for <font color="Olive" title="b3">H</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st <font color="Olive" title="b1">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">G2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b4">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>  st <font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">H</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c6">G1</font>,<font color="Maroon" title="c7">G2</font>  st <font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <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> )</span><br/>

















<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <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></span><br/>

<a NAME="E2:38"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <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> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E32">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <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="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font></span><br/><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
( <font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T76" title="CAT_1:th.76">CAT_1:76</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <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="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</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> <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="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="isocat_1.html#T22" target="_self" title="ISOCAT_1:th.22">Th27</a>, <a class="ref" href="isocat_1.html#T24" target="_self" title="ISOCAT_1:th.24">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c8">H</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> <span class="p2">(<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="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="isocat_1.html#T23" target="_self" title="ISOCAT_1:th.23">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c8">H</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="isocat_1.html#T24" target="_self" title="ISOCAT_1:th.24">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c10">v</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="isocat_1.html#T22" target="_self" title="ISOCAT_1:th.22">Th27</a>, <a class="ref" href="isocat_1.html#T23" target="_self" title="ISOCAT_1:th.23">Th28</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:38"/>
<font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="isocat_1.html#T22" target="_self" title="ISOCAT_1:th.22">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:38"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="isocat_1.html#T22" target="_self" title="ISOCAT_1:th.22">Th27</a>, <a class="ref" href="isocat_1.html#T26" target="_self" title="ISOCAT_1:th.26">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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