<?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" target="_self">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span>;<br/>





<a NAME="E1:66"/><i><font color="Green">E34</font></i>: 
 <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>thus </b><a NAME="E2:66"/>
( not <font color="Maroon">S</font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Maroon">S</font> is <a href="pencil_1.html#V2" target="_self">strong</a> &amp; <font color="Maroon">S</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a> implies  ex <font color="Olive">B</font> being 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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Olive">B</font></span>)</span></span>)</span>  st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <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="Olive">B</font></span>)</span> holds <br/>( <font color="Olive">C</font> is <a href="pencil_1.html#V2" target="_self">strong</a> &amp; <font color="Olive">C</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">S</font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Maroon">S</font> is <a href="pencil_1.html#V2" target="_self">strong</a> &amp; <font color="Maroon">S</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a> )
 ;<br/>

<a NAME="E2:66_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:66_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<a NAME="E7:66_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <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/>
<a NAME="E8:66_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 
<b>by </b><i>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E10:66_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>consider </b><font color="Maroon">CC</font> being  <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E12:66_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font> &amp;  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E13:66_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font> )
 
<b>by </b><i>, <a class="txt" href="pencil_1.html#E10:66_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E14:66_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">CC</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">CC</font> = <font color="Maroon">CC</font> as  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E16:66_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a1</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font></span>)</span> &amp; <font color="Maroon">b1</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a1</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <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">CC</font></span>)</span>
 ;<br/><a NAME="E2:66_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_1_2"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font> )
 <b>by </b><i>, <a class="txt" href="pencil_1.html#E10:66_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E3:66_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:66_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:66_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> =  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font></span>)</span> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E20:66_1"/>
<font color="Maroon">CC</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">CC</font></span>)</span>,<font color="Maroon">p</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:66_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><a NAME="E2:66_1_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">CC</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">CC</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_2_1_1"/>
<font color="Maroon">CC</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><b>hence </b><a NAME="E4:66_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">CC</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">CC</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_2_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><a NAME="E2:66_1_2_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">CC</font> <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/><a NAME="E3:66_1_2_1_2"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:66_1_2_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:66_1_2_1_2_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<a NAME="E4:66_1_2_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_2_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:66_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">CC</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">CC</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">CC</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">CC</font></span>)</span>,<font color="Maroon">p</font> as  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E22:66_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">CC</font> <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/>
<a NAME="E23:66_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">B</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E24:66_1"/><i><font color="Green">E86</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <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>take </b>
<font color="Maroon">B</font>
;<br/>

<a NAME="E25:66_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">f</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E1:66_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E3:66_1_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_3_1"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:66_1_3_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font>
 <b>by </b><i>, <a class="txt" href="pencil_1.html#E10:66_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E3:66_1_3_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E4:66_1_3_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">CC</font>
 <b>by </b><i>, <a class="txt" href="pencil_1.html#E10:66_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E5:66_1_3_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E6:66_1_3_1"/>
<font color="Maroon">f1</font>,<font color="Maroon">a</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , </i>;<br/><b>then consider </b><font color="Maroon">la</font> being   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E8:66_1_3_1"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">la</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, </i>;<br/><b>consider </b><font color="Maroon">La</font> being  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E10:66_1_3_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">la</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">La</font> &amp; <font color="Maroon">La</font> <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">La</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">La</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E11:66_1_3_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">La</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">La</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E12:66_1_3_1"/><b>then </b>
 <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">La</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E13:66_1_3_1"/><b>then </b>
( not <font color="Maroon">La</font> <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">CC</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">La</font> <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">CC</font></span>)</span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, </i>;<br/><b>then consider </b><font color="Maroon">ca</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:66_1_3_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">La</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ca</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="E16:66_1_3_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">La</font> <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/><a NAME="E17:66_1_3_1"/><b>then </b><i><font color="Green">E153</font></i>: 
( <font color="Maroon">a1</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ca</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ca</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D7" target="_self">Def7</a>, <a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E18:66_1_3_1"/>
<font color="Maroon">f1</font>,<font color="Maroon">b</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , </i>;<br/><b>then consider </b><font color="Maroon">lb</font> being   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E20:66_1_3_1"/><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">lb</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, </i>;<br/><b>consider </b><font color="Maroon">Lb</font> being  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E22:66_1_3_1"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">lb</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">Lb</font> &amp; <font color="Maroon">Lb</font> <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">Lb</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">Lb</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E23:66_1_3_1"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">Lb</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">Lb</font> )
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E24:66_1_3_1"/><b>then </b>
 <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">Lb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E25:66_1_3_1"/><b>then </b>
( not <font color="Maroon">Lb</font> <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">CC</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Lb</font> <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">CC</font></span>)</span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, </i>;<br/><b>then consider </b><font color="Maroon">cb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:66_1_3_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">Lb</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">cb</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="E28:66_1_3_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Lb</font> <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/><a NAME="E29:66_1_3_1"/><b>then </b><i><font color="Green">E158</font></i>: 
( <font color="Maroon">b1</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">cb</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">cb</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E30:66_1_3_1"/><b>then </b>
<font color="Maroon">b1</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cb</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E31:66_1_3_1"/><b>then </b>
<font color="Maroon">ca</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cb</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T6" target="_self">Th6</a>, <a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E32:66_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <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">CC</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">cb</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E33:66_1_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E26:66_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_4"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
  <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="66_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:66_1_4_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">e</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<a NAME="E3:66_1_4_1"/><i><font color="Green">E161</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_4_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_4_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_4_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_4_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_4_1_1_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><a NAME="E2:66_1_4_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:66_1_4_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <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="E5:66_1_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <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">e</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="E2:66_1_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:66_1_4"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:66_1_4"/>
<font color="Maroon">f</font> <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">CC</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:66_1_4"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <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">CC</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <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">CC</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E9:66_1_4"/><i><font color="Green">E177</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_4_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_4_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_4_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_4_2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_4_2_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">CC</font>
 ;<br/><div class="add"><a NAME="E2:66_1_4_2_1_2"/><b>then </b>
( not <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:66_1_4_2_1_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c</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="E5:66_1_4_2_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:66_1_4_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:66_1_4_2_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:66_1_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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

<b>let </b><font color="Maroon">SS</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E27:66_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">SS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <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">B</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E28:66_1"/>
<font color="Maroon">SS</font> is <a href="pencil_1.html#V2" target="_self">strong</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>, <font color="Maroon">r</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" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D3" target="_self">PENCIL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_5"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font> )
 ;<br/>

<b>thus </b><a NAME="E2:66_1_5"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_5_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_5_1_1_1"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">r</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">Q1</font> = <font color="Maroon">Q</font>, <font color="Maroon">R1</font> = <font color="Maroon">R</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/><i><font color="Green">E185</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q1</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> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E6:66_1_5_1_1_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">Q1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_5_1_1_2_2"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:66_1_5_1_1_2_2"/><b>then </b>
<font color="Maroon">i</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="E3:66_1_5_1_1_2_2"/><b>then </b><i><font color="Green">E187</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</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="66_1_5_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_5_1_1_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_5_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">Q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_5_1_1_2_2_1_1"/>
<font color="Maroon">Q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_5_1_1_2_2_1_2"/>
<font color="Maroon">Q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><i><font color="Green">E189</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R1</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> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E9:66_1_5_1_1_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_5_1_1_2_4"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:66_1_5_1_1_2_4"/><b>then </b>
<font color="Maroon">i</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="E3:66_1_5_1_1_2_4"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</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="66_1_5_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_5_1_1_2_4_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_4_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_5_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_5_1_1_2_4_1_1"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="pencil_1.html#D7" target="_self">Def7</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_5_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:66_1_5_1_1_2_4_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_5_1_1_2_4_1_2"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:66_1_5_1_1_2"/><b>then </b>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E11:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">Q</font>,<font color="Maroon">R</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i>, , <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_5_1_1_2_5"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:66_1_5_1_1_2_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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/><a NAME="E3:66_1_5_1_1_2_5"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">Q1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E4:66_1_5_1_1_2_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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>hence </b><a NAME="E5:66_1_5_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">L</font> being   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E14:66_1_5_1_1_2"/><i><font color="Green">E195</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">Q</font>,<font color="Maroon">R</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D7" target="_self">Def7</a>, </i>;<br/><b>consider </b><font color="Maroon">L1</font> being  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E16:66_1_5_1_1_2"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L1</font> <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">L1</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L1</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E17:66_1_5_1_1_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E18:66_1_5_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <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/><a NAME="E19:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">Q1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> <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">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E20:66_1_5_1_1_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><a NAME="E21:66_1_5_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <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/><a NAME="E22:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">R1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> <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">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_5_1_1_2_6"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">Q1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> <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">B</font></span>)</span>
 ;<br/><a NAME="E2:66_1_5_1_1_2_6"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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/><a NAME="E3:66_1_5_1_1_2_6"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">Q1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E4:66_1_5_1_1_2_6"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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>hence </b><a NAME="E5:66_1_5_1_1_2_6"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, , <a class="txt" href="pencil_1.html#E12:66_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><a NAME="E24:66_1_5_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">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E25:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
not <font color="Maroon">L1</font> <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">B</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i/>;<br/><a NAME="E26:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L1</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">LI</font> = <font color="Maroon">L1</font> <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">L1</font></span>)</span> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, , </i>;<br/><a NAME="E28:66_1_5_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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/><a NAME="E29:66_1_5_1_1_2"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">Q1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E30:66_1_5_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <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/><a NAME="E31:66_1_5_1_1_2"/><b>then </b>
<font color="Maroon">R1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E32:66_1_5_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">LI</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, <a class="txt" href="pencil_1.html#E12:66_1"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E33:66_1_5_1_1_2"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


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

<b>thus </b><a NAME="E29:66_1"/>
<font color="Maroon">SS</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D2" target="_self">PENCIL_1:def 2</a><br/>

<a NAME="E1:66_1_6"/>
<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:66_1_6"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
;<br/>
<b>assume </b><a NAME="E3:66_1_6"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">SS</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:66_1_6"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">SS</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">SS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:66_1_6"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L</font> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x1</font> = <font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">y1</font> = <font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font>;<br/>
<a NAME="E8:66_1_6"/><i><font color="Green">E223</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E9:66_1_6"/>
<font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_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="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:66_1_6_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>thus </b><a NAME="E2:66_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_1_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_1_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E225</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_1_1_1_1"/>
<font color="Maroon">B</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_1_1_1_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, , <a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">L2</font> = <font color="Maroon">B</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">B</font></span>)</span>,<font color="Maroon">L1</font> as  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:66_1_6"/><i><font color="Green">E228</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <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/>
<a NAME="E12:66_1_6"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">L2</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E13:66_1_6"/><b>then </b><i><font color="Green">E230</font></i>: 
 <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:66_1_6"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <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/>
<a NAME="E15:66_1_6"/><b>then </b>
<font color="Maroon">L2</font> <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">L2</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L3</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">L2</font> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E17:66_1_6"/><i><font color="Green">E232</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_6_2"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_2_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_2_1_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_2_1_1"/><b>then </b><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_2_1_1"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_2_1_2"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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/><b>hence </b><a NAME="E3:66_1_6_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E19:66_1_6"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_6_3"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_3_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_3_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_3_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_3_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_3_1_2"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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/><b>hence </b><a NAME="E3:66_1_6_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:66_1_6"/><b>then </b>
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#D8" target="_self">Def8</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E22:66_1_6"/><b>then </b><i><font color="Green">E240</font></i>: 
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D9" target="_self">Def9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:66_1_6"/><i><font color="Green">E241</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_6_4"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_4_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_4_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_4_1_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_4_1_1"/><b>then </b><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:66_1_6_4_1_1"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:66_1_6_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_4_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_4_1_2"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:66_1_6"/><b>then </b><i><font color="Green">E246</font></i>: 
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_6_5"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_5_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_5_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_5_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_5_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_5_1_2"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E27:66_1_6"/><b>then </b>
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#D11" target="_self">Def11</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E28:66_1_6"/><b>then </b>
<font color="Maroon">a1</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">B</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E29:66_1_6"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">L3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, , </i>;<br/>
<a NAME="E30:66_1_6"/><b>then </b><i><font color="Green">E249</font></i>: 
<font color="Maroon">L3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E31:66_1_6"/>
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:66_1_6_6"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_1_6_6"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</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">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:66_1_6_6"/><i><font color="Green">E252</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</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> <font color="Maroon">I</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="txt" href="pencil_1.html#E12:66_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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/>
<a NAME="E7:66_1_6_6"/><i><font color="Green">E253</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_6_6_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_6_6_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_6_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_6_6_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, , <a class="txt" href="pencil_1.html#E17:66_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_6_6_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_6_6_1_1_2"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_1_6_6_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_1_6_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:66_1_6_6"/><b>then </b>
<font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L3</font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E16:66_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E10:66_1_6_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</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">B</font></span>)</span>,<font color="Maroon">e</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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <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">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pencil_1.html#T7" target="_self">Th7</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E11:66_1_6_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>


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


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


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

<b>given </b><font color="Maroon">B</font> being  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><a NAME="E3:66"/><i><font color="Green">E256</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>  st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <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">B</font></span>)</span> holds <br/>( <font color="Olive">C</font> is <a href="pencil_1.html#V2" target="_self">strong</a> &amp; <font color="Olive">C</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a> ) ) )
 ;<br/>

<b>thus </b><a NAME="E4:66"/>
not <font color="Maroon">S</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i/>;<br/>

<a NAME="E5:66"/><i><font color="Green">E257</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <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>thus </b><a NAME="E6:66"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V2" target="_self">strong</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D3" target="_self">PENCIL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:66_2_1_2"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <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> &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <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> holds <br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E4:66_2_1_2"/>
 <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">x1</font> = <font color="Maroon">x1</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>consider </b><font color="Maroon">y1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:66_2_1_2"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <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> &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <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> holds <br/><font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:66_2_1_2"/>
 <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">y1</font> = <font color="Maroon">y1</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green">E262</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1_2_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">x1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <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">B</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <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>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_2_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_2_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:66_2_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E11:66_2_1_2"/>
<font color="Maroon">x1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#T3" target="_self">Th3</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">x1i</font> = <font color="Maroon">x1</font> <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">B</font></span>)</span> as   <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" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/><a NAME="E13:66_2_1_2"/>
<font color="Maroon">y1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#T3" target="_self">Th3</a>, <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a></i>;<br/><b>then reconsider </b><font color="Maroon">y1i</font> = <font color="Maroon">y1</font> <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">B</font></span>)</span> as   <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" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/><a NAME="E15:66_2_1_2"/>
<font color="Maroon">B</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E16:66_2_1_2"/><b>then </b>
<font color="Maroon">B</font> <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">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">B</font> <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">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/><a NAME="E18:66_2_1_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">x1i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E19:66_2_1_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">y1i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E20:66_2_1_2"/>
<font color="Maroon">C</font> is <a href="pencil_1.html#V2" target="_self">strong</a>
 <b>by </b><i/>;<br/><a NAME="E21:66_2_1_2"/><b>then </b>
<font color="Maroon">x1i</font>,<font color="Maroon">y1i</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E23:66_2_1_2"/><i><font color="Green">E268</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x1i</font>,<font color="Maroon">y1i</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E24:66_2_1_2"/><i><font color="Green">E269</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></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/><a NAME="E25:66_2_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> holds <br/>( not <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="realset1.html#V1">trivial</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E1:66_2_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:66_2_1_2_2"/>
( not <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="realset1.html#V1">trivial</a> )
 ;<br/>


</div><b>end;</b></div><a NAME="E26:66_2_1_2"/><b>then </b><i><font color="Green">E270</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font> is <a href="pencil_1.html#V16" target="_self">Segre-like</a>
 <b>by </b><i/>;<br/><a NAME="E27:66_2_1_2"/>
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:66_2_1_2_3"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>thus </b><a NAME="E3:66_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_3_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_3_1_1_1"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:66_2_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:66_2_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E5:66_2_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_3_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E273</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>
;<br/><a NAME="E3:66_2_1_2_3_1_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E4:66_2_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/><a NAME="E5:66_2_1_2_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:66_2_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div><a NAME="E28:66_2_1_2"/><b>then </b><i><font color="Green">E274</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E29:66_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</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">B</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>then reconsider </b><font color="Maroon">L</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pencil_1.html#T1" target="_self">Th1</a></i>;<br/><i><font color="Green">E275</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</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"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E32:66_2_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> holds <br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_2_1_2_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span>
 ;<br/>

<a NAME="E2:66_2_1_2_5"/><b>then </b><i><font color="Green">E276</font></i>: 
<font color="Maroon">a</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_5_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_5_1_1"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_2_5_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_5_1_2"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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/><a NAME="E3:66_2_1_2_5_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <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><a NAME="E33:66_2_1_2"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D3" target="_self">Def3</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><i><font color="Green">E280</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y1</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"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E35:66_2_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> holds <br/><font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_2_1_2_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span>
 ;<br/>

<a NAME="E2:66_2_1_2_7"/><b>then </b><i><font color="Green">E281</font></i>: 
<font color="Maroon">a</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_7_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_2_7_1_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_2_7_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_2_7_1_2"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_7_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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/><a NAME="E3:66_2_1_2_7_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D6" target="_self">Def6</a>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:66_2_1_2_7_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_2_7_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> <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">B</font></span>)</span>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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><a NAME="E36:66_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E37:66_2_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><b>hence </b><a NAME="E38:66_2_1_2"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1" target="_self">are_collinear</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E7:66"/>
<font color="Maroon">S</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D2" target="_self">PENCIL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:66_3"/><i><font color="Green">E284</font></i>: 
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_3"/><i><font color="Green">E285</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:66_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L</font> being  <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:66_3"/><i><font color="Green">E286</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L</font> &amp;  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:66_3"/>
<font color="Maroon">l</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:66_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E10:66_3"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">l</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as  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#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<a NAME="E12:66_3"/><i><font color="Green">E287</font></i>: 
(  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 
<b>by </b><i>, , , </i>;<br/>
<i><font color="Green">E288</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</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> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E14:66_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_3_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_1"><b>suppose </b></a><a NAME="E1:66_3_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_3_2_1_1"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2"><b>suppose </b></a><a NAME="E1:66_3_2_1_2"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:66_3_2_1_2"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:66_3_2_1_2"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:66_3_2_1_2"/><b>then </b>
not <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i/>;<br/><a NAME="E6:66_3_2_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">Li</font> = <font color="Maroon">L</font> <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">B</font></span>)</span> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E8:66_3_2_1_2"/>
<font color="Maroon">B</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E9:66_3_2_1_2"/><b>then </b>
<font color="Maroon">B</font> <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">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">B</font> <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">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/><a NAME="E11:66_3_2_1_2"/><i><font color="Green">E292</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green">E293</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_3_2_1_2_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">a1</font> <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">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <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">B</font></span>)</span>
 ;<br/><a NAME="E2:66_3_2_1_2_1"/><i><font color="Green">E295</font></i>: 
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">a1</font> holds <br/><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_2_1_2_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> ;<br/>

<b>assume </b><a NAME="E1:66_3_2_1_2_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_3_2_1_2_1_1_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_3_2_1_2_1_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_3_2_1_2_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_3_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_3_2_1_2_1_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_3_2_1_2_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div> <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</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> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:66_3_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E13:66_3_2_1_2"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:66_3_2_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
( <font color="Maroon">a1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <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">B</font></span>)</span> &amp; <font color="Maroon">b1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <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">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#T6" target="_self">Th6</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E15:66_3_2_1_2"/>
( <font color="Maroon">a1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <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">B</font></span>)</span> &amp; <font color="Maroon">b1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <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">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pencil_1.html#D1" target="_self">Def1</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E16:66_3_2_1_2"/><b>then </b>
( <font color="Maroon">a1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b1</font> <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">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Li</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:66_3_2_1_2"/><b>then </b><i><font color="Green">E297</font></i>: 
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Li</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E18:66_3_2_1_2"/>
<font color="Maroon">C</font> is <a href="pencil_1.html#V1" target="_self">closed_under_lines</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E19:66_3_2_1_2"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E15:66_3"/>
<font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="pencil_1.html#T6" target="_self">Th6</a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>


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


</div>
