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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">B1</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>, <font color="Maroon">B2</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font>
 ;<br/>

<b>thus </b><a NAME="E2:23"/>
( <font color="Maroon">B1</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">B2</font> implies for <font color="Olive">b1</font>, <font color="Olive">b2</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> holds <br/>(  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> &amp;  ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> &amp; ( 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> <font color="Olive">r</font> holds <br/><font color="Olive">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">c1</font>, <font color="Olive">c2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Olive">r</font></span>)</span>  st <font color="Olive">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">c1</font>,<font color="Olive">c2</font> <a href="pencil_1.html#R1">are_collinear</a>  ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">B1</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">B2</font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> holds <br/> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="pencil_1.html#R1">are_collinear</a>  )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">b1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>, <font color="Maroon">b2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E3:23_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> )
 ;<br/>

<b>consider </b><font color="Maroon">L1</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">B1</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">indx</a> <font color="Maroon">L1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">L2</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:23_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E8:23_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">L1</font>
 
<b>by </b><i>, , <a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/>
<a NAME="E9:23_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, , <a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/>
<b>thus </b><a NAME="E10:23_1"/><i><font color="Green">E53</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E2:23_1_1"/><b>then </b>
( not <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c2</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/>
<b>consider </b><font color="Maroon">p0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:23_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">p0</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">bl</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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:23_1_1"/><i><font color="Green">E70</font></i>: 
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">bl</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E11:23_1_1"/>
<font color="Maroon">bl</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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:23_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">bl</font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E13:23_1_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:23_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T3">PENCIL_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> 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">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<i><font color="Green">E74</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> <font color="Maroon">b1</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="23_1_1_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="E1:23_1_1_2"/><i><font color="Green">E75</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">x1</font>
 ;<br/><a NAME="E2:23_1_1_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="E3:23_1_1_2"/><b>then </b><i><font color="Green">E102</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">p</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="23_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1"/><b>then </b>
<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">a</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:23_1_1_2_1_1"/><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>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_2_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:23_1_1_2_1_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; ( for <font color="Olive">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">b1</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <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="E4:23_1_1_2_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">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_2_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> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:23_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E23:23_1_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1">are_collinear</a>  )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E3:23_1_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:23_1_1_3_2"/><i><font color="Green">E108</font></i>: 
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i0</font></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="pencil_1.html#R1">are_collinear</a>  ) ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 <b>by </b><i>, </i>;<br/><i><font color="Green">E109</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> <font color="Maroon">b1</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="23_1_1_3_2_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="E1:23_1_1_3_2_2"/><i><font color="Green">E110</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">y1</font>
 ;<br/><a NAME="E2:23_1_1_3_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="E3:23_1_1_3_2_2"/><b>then </b><i><font color="Green">E111</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">p</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">i5</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <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="23_1_1_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_2_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>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> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:23_1_1_3_2_2_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is    <a href="subset_1.html#M1">Element</a> of <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="pboole.html#D17">PBOOLE:def 17</a></i>;<br/><a NAME="E4:23_1_1_3_2_2_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:23_1_1_3_2_2_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E6:23_1_1_3_2_2_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</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">.</a> <font color="Maroon">i1</font></span>)</span>
 ;<br/><a NAME="E7:23_1_1_3_2_2_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_3_2_2_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_2_2_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:23_1_1_3_2_2_1_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; ( for <font color="Olive">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">b1</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <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="E4:23_1_1_3_2_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</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="E5:23_1_1_3_2_2_1_2"/><b>then </b><i><font color="Green">E141</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">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:23_1_1_3_2_2_1_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> &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> <font color="Maroon">b2</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <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="E8:23_1_1_3_2_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</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="E9:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</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">c2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E11:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:23_1_1_3_2_2_1_2"/><b>then </b>
<font color="Maroon">i0</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E14:23_1_1_3_2_2_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i5</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E15:23_1_1_3_2_2_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">p</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="E16:23_1_1_3_2_2_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</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_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:23_1_1_3_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E7:23_1_1_3_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_3_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E11:23_1"/>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> &amp; ( 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> <font color="Olive">r</font> holds <br/><font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">c1</font>, <font color="Olive">c2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Olive">r</font></span>)</span>  st <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">c1</font>,<font color="Olive">c2</font> <a href="pencil_1.html#R1">are_collinear</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:23_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E6:23_1_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1">are_collinear</a>  )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:23_1_2_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:23_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E10:23_1_2"/><i><font color="Green">E147</font></i>: 
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="pencil_1.html#R1">are_collinear</a>  ) ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><i><font color="Green">E149</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> <font color="Maroon">b1</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="23_1_2_2_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="E1:23_1_2_2_2"/><i><font color="Green">E150</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">b1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <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="23_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_2_2_1"/>
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">o1</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="23_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_2_2_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_2_2_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:23_1_2_2_2_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> is    <a href="subset_1.html#M1">Element</a> of <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">o1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/><a NAME="E4:23_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">o1</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:23_1_2_2_2_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_2_2_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_2_2_1_2"/><b>then </b>
( <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> is <a href="realset1.html#V1">trivial</a> &amp; not <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></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="E4:23_1_2_2_2_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</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="E5:23_1_2_2_2_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
 <b>by </b><i>, , , <a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>then </b><font color="Maroon">c</font> = 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E7:23_1_2_2_2_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:23_1_2_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E5:23_1_2_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:23_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:23_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/>

<b>thus </b><a NAME="E13:23_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> <font color="Maroon">r</font> holds <br/><font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_3"><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="E1:23_1_2_3"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_3_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_3_1_1"/><b>then </b>
( <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_2_3_1_1"/>
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_3_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:23_1_2_3_1_2"/>
( <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="realset1.html#V1">trivial</a> &amp; not <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></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="E4:23_1_2_3_1_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">b1</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="E5:23_1_2_3_1_2"/>
( <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="realset1.html#V1">trivial</a> &amp; not <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, <a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:23_1_2_3_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">b2</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">d</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="E8:23_1_2_3_1_2"/>
 <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="E9:23_1_2_3_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> <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="card_3.html#T18">CARD_3:18</a></i>;<br/><b>then </b><i><font color="Green">E161</font></i>: <font color="Maroon">c</font> = 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><a NAME="E11:23_1_2_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</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="E12:23_1_2_3_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</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/><b>hence </b><a NAME="E13:23_1_2_3_1_2"/>
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:23_1"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="pencil_3.html#E3:23_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">c1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>, <font color="Maroon">c2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>;<br/>

<b>assume </b><a NAME="E14:23_1_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c2</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E15:23_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="E16:23_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E17:23_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:23_1_2"/>
 <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="E19:23_1_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E20:23_1_2"/><b>then </b>
<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E21:23_1_2"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a></i>;<br/>


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


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

<b>assume </b><a NAME="E3:23"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">b1</font>, <font color="Olive">b2</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> holds <br/>(  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> &amp;  ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> &amp; ( 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> <font color="Olive">r</font> holds <br/><font color="Olive">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">c1</font>, <font color="Olive">c2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Olive">r</font></span>)</span>  st <font color="Olive">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">c1</font>,<font color="Olive">c2</font> <a href="pencil_1.html#R1">are_collinear</a>  ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">L1</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">B1</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">indx</a> <font color="Maroon">L1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">L2</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:23"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green">E167</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E10:23"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font> &amp; ( 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> <font color="Maroon">r</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">c1</font>, <font color="Olive">c2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>  st <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">c2</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">c1</font>,<font color="Olive">c2</font> <a href="pencil_1.html#R1">are_collinear</a>  ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E11:23"/>
( not <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:23"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</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="E14:23"/>
<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>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:23"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</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">r</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E16:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c1</font></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">.</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c1</font> 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">.</a> <font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E19:23"/>
( not <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i>, , <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:23"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c2</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="E22:23"/>
<font color="Maroon">L2</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="E23:23"/><b>then </b>
<font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</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">r</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E24:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E25:23"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c2</font></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">.</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">c2</font> 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">.</a> <font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<div><i><font color="Green">E171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 ;<br/><i><font color="Green">E173</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</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">L2</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="23_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <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="23_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_2_1"/>
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">s1</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="23_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_2_2_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_2_2_1_1"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_2_2_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_2_2_1_2"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:23_2"/><b>then </b>
<font color="Maroon">L1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E5:23_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:23_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E7:23_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E28:23"/>
<font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="pencil_1.html#R1">are_collinear</a> 
 
<b>by </b><i>, , <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">bb</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">.</a> <font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E30:23"/><i><font color="Green">E175</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">bb</font>
 <b>by </b><i>, <a class="ref" href="pencil_1.html#D1">PENCIL_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_3.html#D2" target="_self">PENCIL_3:def 2</a><br/>

<b>assume </b><a NAME="E31:23"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E34:23"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> &amp; ( for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> holds <br/><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x1</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">r</font>,<font color="Maroon">c2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/>
<i><font color="Green">E179</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> <font color="Maroon">L2</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="23_4"><b>now </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="E1:23_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <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="23_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">a1</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="23_4_1_1"><b>suppose </b></a><a NAME="E1:23_4_1_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</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="E3:23_4_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E4:23_4_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> <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_2"><b>suppose </b></a><a NAME="E1:23_4_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_2"/><i><font color="Green">E182</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</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="E3:23_4_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E4:23_4_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:23_4_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> <font color="Maroon">L2</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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E39:23"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:23"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>

<b>reconsider </b><font color="Maroon">B</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> <font color="Maroon">r</font>,<font color="Maroon">bb</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">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E184</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_6"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</font>
 ;<br/><a NAME="E2:23_6"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">s</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="E3:23_6"/><b>then </b><i><font color="Green">E187</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_6_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">s</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="23_6_1_1"><b>suppose </b></a><a NAME="E1:23_6_1_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_6_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:23_6_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:23_6_1_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:23_6_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E6:23_6_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <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="23_6_1_2"><b>suppose </b></a><a NAME="E1:23_6_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_6_1_2"/><b>then </b>
<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">s</font> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:23_6_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">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:23_6_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E43:23"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green">E191</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_8"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y1</font>
 ;<br/><a NAME="E2:23_8"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">s</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="E3:23_8"/><b>then </b><i><font color="Green">E193</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="pzfmisc1.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</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="23_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_8_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">s</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="23_8_1_1"><b>suppose </b></a><a NAME="E1:23_8_1_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_8_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:23_8_1_1"/>
<font color="Maroon">bb</font> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E4:23_8_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <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="23_8_1_2"><b>suppose </b></a><a NAME="E1:23_8_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:23_8_1_2"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:23_8_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="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E4:23_8_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">s</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:23_8_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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> <font color="Maroon">r</font>,<font color="Maroon">bb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E46:23"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E47:23"/>
<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">B</font>
 
<b>by </b><i>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>hence </b><a NAME="E48:23"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i><a class="ref" href="pencil_1.html#D1">PENCIL_1:def 1</a></i>;<br/>


</div>
