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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_2_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V9">semi-functional</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="yellow18.html#V3">set-id-inheriting</a> )
 <b>and </b><br/><a NAME="E2:8_2_1"/><i><font color="Green">E7</font></i>: 
for <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> holds <font color="Olive">b<sub>1</sub></font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 <b>and </b><br/><a NAME="E3:8_2_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D4" target="_self">YELLOW21:def 4</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a> </span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM9">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:8_2_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</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">H<sub>1</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="E6:8_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow18.html#V2">para-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7">YELLOW18:def 7</a><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">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="yellow21.html#E2:8_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E2:8_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:8_2_1_1"/><b>then </b>
( <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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E5:8_2_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E4:8_2_1_1"/><b>then </b>
( <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="tarski.html#R1">c=</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:8_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="orders_3.html#T11">ORDERS_3:11</a></i>;<br/>
<b>hence </b><a NAME="E5:8_2_1_1"/>
<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="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:8_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow18.html#V4">concrete</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:8_2_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="yellow18.html#D11">YELLOW18:def 11</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D3" target="_self">YELLOW21:def 3</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="yellow21.html#E2:8_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:8_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E10:8_2_1"/>
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <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>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <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>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:8_2_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E12:8_2_1"/><i><font color="Green">E10</font></i>: 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E13:8_2_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E14:8_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="orders_3.html#D6">ORDERS_3:def 6</a></i>;<br/>
<b>thus </b> <a href="yellow18.html#K3">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="yellow18.html#K3">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="yellow21.html#E1:8_2_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow21.html#E12:8_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow18.html#D10">YELLOW18:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="yellow21.html#E13:8_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>
;<br/>


</div>
