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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> be   <a href="altcat_1.html#NM4" title="ALTCAT_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> being <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0: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="b3">G1</font>, <font color="Olive" title="b4">G2</font> being <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0: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="b5">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font><br/> for <font color="Olive" title="b6">f</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b4">G2</font>  st <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b4">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b4">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b2">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b6">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Olive" title="b5">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b4">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b2">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0: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">G1</font>, <font color="Olive" title="b2">G2</font> being <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0: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">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><br/> for <font color="Olive" title="b4">f</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Olive" title="b2">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b4">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Olive" title="b3">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Olive" title="b2">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0: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">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>  st <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b2">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Olive" title="b1">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c8">e</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">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">f</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>  st <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b1">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Maroon" title="c8">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c9">f</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  implies ( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Maroon" title="c8">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> ) )</span><br/>

















<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</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">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Maroon" title="c8">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/>

<a NAME="E2:55"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
( <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font> &amp; <font color="Maroon" title="c6">G1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> )
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="functor3.html#T35" target="_self" title="FUNCTOR3:th.35">Th35</a>, <a class="ref" href="functor3.html#T36" target="_self" title="FUNCTOR3:th.36">Th36</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E35">A5</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c5">F2</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="functor3.html#T35" target="_self" title="FUNCTOR3:th.35">Th35</a>, <a class="ref" href="functor3.html#T36" target="_self" title="FUNCTOR3:th.36">Th36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">sF2</font> = <font color="Maroon" title="c9">f</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c5">F2</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <b>by </b><i><a class="txt" href="functor3.html#E2:55"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="functor3.html#T29" target="_self" title="FUNCTOR3:th.29">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">G1t</font> = <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">e</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <b>by </b><i><a class="txt" href="functor3.html#E2:55"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="functor3.html#T28" target="_self" title="FUNCTOR3:th.28">Th28</a></i>;<br/>
<a NAME="E8:55"/>
<font color="Maroon" title="c10">sF2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c11">G1t</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="functor3.html#E3:55"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="functor3.html#E5:55"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="functor3.html#T34" target="_self" title="FUNCTOR3:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E9:55"/>
( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Maroon" title="c8">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 <b>by </b><i><a class="txt" href="functor3.html#E3:55"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="functor3.html#E4:55"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="functor3.html#T33" target="_self" title="FUNCTOR3:th.33">Th33</a>, <a class="ref" href="functor2.html#D8" title="FUNCTOR2:def.8">FUNCTOR2:def 8</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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