<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E1:26_1_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="functor3.html#E1:26_1_1"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E4:26_1_1"/>
<font color="Maroon">c<sub>9</sub></font> is    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="functor3.html#E1:26"><i><font color="Green">E9</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></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">c<sub>10</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:26_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>]
 
<b>by </b><i><a class="txt" href="functor3.html#E3:26_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E3:26_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor3.html#E2:26_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>4</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E6:26_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>]
 
<b>by </b><i><a class="txt" href="functor3.html#E3:26_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:26_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>


</div>
