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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>thus </b><a NAME="E1:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V16" target="_self">Segre-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D20" target="_self">PENCIL_1:def 20</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:64_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pencil_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:64"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D16" target="_self">PENCIL_1:def 16</a><br/>

<a NAME="E1:64_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:64_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E3:64_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_2"/>
not <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>


</div><b>end;</b></div>


</div>
