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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>;<br/>
<a NAME="E1:10_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>3</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>4</sub></font> ) </span>}</span>  iff <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:10_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:10_1_1_1_1"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:10_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>and </b><br/><a NAME="E5:10_1_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E6:10_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E3:10_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:10_1_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  iff <font color="Maroon">c<sub>5</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:10_1_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:10_1_1_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E5:10_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:10_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:10_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="afproj.html#E3:10_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:10_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>, <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:10_1_1_3"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/><a NAME="E4:10_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:10_1_1_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="aff_4.html#T57">AFF_4:57</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:10_1_1_3"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:10_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E7:10_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_1_1_4"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:10_1_1_4"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:10_1_1_4"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_4"><i><font color="Green">E10</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:10_1_1_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_4"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E7:10_1_1_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="afproj.html#E2:10_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="afproj.html#E4:10_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="afproj.html#E6:10_1_1_4"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:10_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:10_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="afproj.html#E6:10_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="aff_4.html#T58">AFF_4:58</a></i>;<br/><b>hence </b><a NAME="E9:10_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:10_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="afproj.html#E6:10_1_1_4"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:10_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_1_1_5"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:10_1_1_5"/><i><font color="Green">E12</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:10_1_1_5"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_5"><i><font color="Green">E11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:10_1_1_5"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_5"><i><font color="Green">E11</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:10_1_1_5"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1_5"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E9:10_1_1_5"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:10_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="afproj.html#E2:10_1_1_5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="afproj.html#E4:10_1_1_5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="afproj.html#E6:10_1_1_5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="afproj.html#E8:10_1_1_5"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E10:10_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:10_1_1_5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="afproj.html#E6:10_1_1_5"><i><font color="Green">E14</font></i></a>, <a class="txt" href="afproj.html#E8:10_1_1_5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_4.html#T61">AFF_4:61</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:10_1_1_5"><i><font color="Green">E13</font></i></a>, <a class="txt" href="afproj.html#E8:10_1_1_5"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:10_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:10_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E7:10_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="afproj.html#E9:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


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