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<b>let </b><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">A</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="c2">A</font>,<font color="Maroon" title="c1">B</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="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font><br/> for <font color="Olive" title="b4">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b4">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span><br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">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="c2">A</font>,<font color="Maroon" title="c1">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">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">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">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span><br/><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">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">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span><br/><b>let </b><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</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>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  )</span><br/>











<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span><br/>

<a NAME="E2:57"/><b>then </b>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o</font>
<b>by </b><i><a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
<b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="functor3.html#D6" target="_self" title="FUNCTOR3:def.6">Def6</a></i>
;<br/> <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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