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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>









<b>assume </b><a NAME="E1:58"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:58"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:58_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#D5" target="_self">Def5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K5">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="functor3.html#E2:58"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor3.html#E4:58"><i><font color="Green">E37</font></i></a>, <a class="ref" href="functor2.html#D8">FUNCTOR2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E3:58"><i><font color="Green">E36</font></i></a>, <a class="txt" href="functor3.html#E5:58"><i><font color="Green">E38</font></i></a>, <a class="ref" href="functor2.html#D5">FUNCTOR2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="altcat_3.html#K1">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#T38" target="_self">Th38</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:58_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor2.html#K4">idt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="functor2.html#T6">FUNCTOR2:6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:58"/>
<font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#K5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="pboole.html#R6">=</a>  <a href="functor2.html#K4">idt</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="functor2.html#T5">FUNCTOR2:5</a></i>;<br/>


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