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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_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>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font>, <font color="Olive" title="b4">F3</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="b5">t</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font><br/>  for <font color="Olive" title="b6">t1</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font>  st <font color="Olive" title="b1">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F1</font> &amp; <font color="Olive" title="b2">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> &amp; <font color="Olive" title="b3">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b4">F3</font> holds <br/> for <font color="Olive" title="b7">t3</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F2</font>,<font color="Olive" title="b4">F3</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b7">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b6">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b5">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b7">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b5">t</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font>, <font color="Maroon" title="c6">F3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">F1</font><br/>  for <font color="Olive" title="b2">t1</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>  st <font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <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="c5">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F3</font> holds <br/> for <font color="Olive" title="b3">t3</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c6">F3</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b2">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">t</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">F1</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">t1</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>  st <font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <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="c5">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F3</font> holds <br/> for <font color="Olive" title="b2">t3</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c6">F3</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">t1</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</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="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <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="c5">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F3</font> implies  for <font color="Olive" title="b1">t3</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c6">F3</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<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="c6">F3</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">t3</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c6">F3</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span></span><br/>

<a NAME="E4:23"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="functor2.html#T8" target="_self" title="FUNCTOR2:th.8">Th10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">t3</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c6">F3</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="c9">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span></span><br/>

<a NAME="E5:23"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="functor2.html#D6" target="_self" title="FUNCTOR2:def.6">Def6</a></span>;<br/>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
( <font color="Maroon" title="c3">F</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="functor2.html#D6" target="_self" title="FUNCTOR2:def.6">Def6</a></span>;<br/>
<a NAME="E7:23"/>
<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">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="functor2.html#T8" target="_self" title="FUNCTOR2:th.8">Th10</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c7">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="functor2.html#D8" target="_self" title="FUNCTOR2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t3</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c8">t1</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c7">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="functor2.html#D8" target="_self" title="FUNCTOR2:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c9">t3</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="functor2.html#T6" target="_self" title="FUNCTOR2:th.6">Th8</a></span>
<br/>.= 
<font color="Maroon" title="c9">t3</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="functor2.html#D8" target="_self" title="FUNCTOR2:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c9">t3</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t1</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c7">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="functor2.html#D8" target="_self" title="FUNCTOR2:def.8">Def8</a></span>
;<br/> <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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