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<div class="add">

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">o</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">o</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">o</font></span>)</span> );<br/>
<a NAME="E1:26_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">s</font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E3:26_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">s</font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">IT</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E4:26_1_1"/>
<font color="Maroon">IT</font> is    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_1_1_2"/>
<font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D2">FUNCTOR2:def 2</a><br/>

<b>let </b><font color="Maroon">o</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:26_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:26_1_1_2"/>
( <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">G2</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1_2"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> is   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F</font> ;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E6:26_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:26_1_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 ;<br/>


</div>
