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

<b>let </b><font color="Maroon">I</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">A</font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">b1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>





<b>set </b><font color="Maroon">i</font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">bb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:18"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">bf</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; ( 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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> holds <br/><font color="Maroon">bf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b><font color="Maroon">p</font> = <font color="Maroon">bb</font>;<br/>

<b>thus </b><a NAME="E7:18"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 <b>by </b><i/>;<br/>

<b>set </b><font color="Maroon">bbx</font> = <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<b>thus </b><a NAME="E8:18"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:18_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><a NAME="E2:18_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:18_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:18_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:18_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:18_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> holds <br/><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green">E70</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> = 
<font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E2:18_1_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E4:18_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_3_1"/>
( <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_3_1_1"><b>suppose </b></a><a NAME="E1:18_1_3_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:18_1_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_1_3_1_1"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_3_1_2"><b>suppose </b></a><a NAME="E1:18_1_3_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_3_1_2"/><b>then </b>
( not <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18_1_3_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E5:18_1_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E6:18_1_3_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:18_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E8:18_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E9:18_1_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:18_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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


</div>
