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

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>

<b>thus </b><a NAME="E3:42_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pencil_1.html#V16" target="_self">Segre-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_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="pencil_1.html#D20" target="_self">PENCIL_1:def 20</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:42_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:42_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
not <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>
<b>assume </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E5:42_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/>
<a NAME="E6:42_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E2:42_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:42_1_1_1"/>
( not <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:42_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pencil_1.html#E5:42_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pencil_1.html#D16" target="_self">Def16</a></i>;<br/>


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

<b>thus </b><a NAME="E4:42_1_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>; <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="E1:42_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <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="E2:42_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E3:42_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="pzfmisc1.html#K1">}</a></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/>
<a NAME="E4:42_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E5:42_1_1_2"/>
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:42_1_1_2"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:42_1_1_2"/>
not <font color="Maroon">c<sub>5</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>


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


</div>
