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<div class="add">

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:22_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:22_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a><br/>

<b>thus </b><a NAME="E1:22_1_1"/>
not <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E1:22"><i><font color="Green">E6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:22_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:22_1"><i><font color="Green">E8</font></i></a></i>;<br/><b>hence </b><a NAME="E2:22_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:22"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E3:22_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="functor2.html#D2">FUNCTOR2:def 2</a></i>;<br/>
<div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:22_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:22_1"><i><font color="Green">E8</font></i></a></i>;<br/><b>hence </b><a NAME="E2:22_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E3:22_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:22_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_1_4"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E3:22_1"><i><font color="Green">E9</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a><br/>

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E2:22_1_4"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>) )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:22_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_1_4"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:22"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:22_1_5"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E6:22_1"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E2:22_1_5"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:22"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:22_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:22_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:22_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>() st <br/>( <font color="Maroon">F<sub>3</sub></font>() <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 <b>by </b><i><a class="txt" href="yellow18.html#E6:22_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="yellow18.html#E7:22_1"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
