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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="c2">B</font>,<font color="Maroon" title="c3">C</font><br/> for <font color="Olive" title="b3">H1</font>, <font color="Olive" title="b4">H2</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="b5">u</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">H1</font>,<font color="Olive" title="b4">H2</font>  st <font color="Olive" title="b3">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b4">H2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Olive" title="b2">G</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="b5">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">F</font></span>)</span></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="c2">B</font>,<font color="Maroon" title="c3">C</font><br/> for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</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="b4">u</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">H1</font>,<font color="Olive" title="b3">H2</font>  st <font color="Olive" title="b2">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">H2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Olive" title="b1">G</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="b4">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span></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="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"> for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</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">u</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font>  st <font color="Olive" title="b1">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">H2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">H1</font>, <font color="Maroon" title="c8">H2</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">u</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</font>  st <font color="Maroon" title="c7">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">H2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">H1</font>,<font color="Maroon" title="c8">H2</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">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">H2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span> )</span><br/>

















<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c7">H1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">H2</font>
 ; <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">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span><br/>

<a NAME="E2:37"/><b>then </b>
<font color="Maroon" title="c7">H1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">H2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">H1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <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="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c7">H1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</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="c7">H1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c8">H2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</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">H2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <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="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</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="c7">H1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</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">H2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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>
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c9">u</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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>
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c9">u</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</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>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="isocat_1.html#T29" target="_self" title="ISOCAT_1:th.29">Th29</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>hence </b><a NAME="E7:37"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <font color="Maroon" title="c6">G</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="c9">u</font> <a href="isocat_1.html#K5" title="ISOCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="isocat_1.html#T31" target="_self" title="ISOCAT_1:th.31">Th31</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>
