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

<b>set </b><font color="Maroon" title="c2">R1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c3">AFP</font> =  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>;<br/>
<b>set </b><font color="Maroon" title="c4">AFP2</font> = <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>;<br/>
<a NAME="E1:9_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> holds <br/>( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">X</font>,<font color="Olive" title="b4">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b3">X</font>, <font color="Olive" title="b4">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b3">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b4">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b4">Y</font> ) </span>}</span>  iff <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c6">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> implies ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  iff <font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font> ) )</span><br/>



<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c5">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c6">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  iff <font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c7">X'</font>, <font color="Maroon" title="c8">Y'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:9_1_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">X'</font>,<font color="Maroon" title="c8">Y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:9_1_1_1_1"/>
( <font color="Maroon" title="c7">X'</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c8">Y'</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>and </b><br/><a NAME="E5:9_1_1_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c7">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">Y'</font>
 ;<br/><a NAME="E6:9_1_1_1_1"/>
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X'</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y'</font> )
 <b>by </b><i><a class="txt" href="#E3:9_1_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="#E5:9_1_1_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  iff <font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_1"><i><font color="Green" title="E6">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )</span><br/><b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><b>then consider </b><font color="Maroon" title="c6">X</font>, <font color="Maroon" title="c7">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:9_1_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:9_1_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
( <font color="Maroon" title="c6">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c7">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Maroon" title="c6">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">Y</font> )
 ;<br/><a NAME="E5:9_1_1_2"/>
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1_2"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E6:9_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:9_1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:9_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span>,<span class="p2">(<span class="default"><a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">R2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>, <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )</span><br/><b>assume </b><a NAME="E1:9_1_1_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font></span><br/><b>then consider </b><font color="Maroon" title="c7">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:9_1_1_3"/><i><font color="Green" title="E6">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c7">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 ;<br/><a NAME="E4:9_1_1_3"/>
<font color="Maroon" title="c7">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="#E3:9_1_1_3"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="aff_4.html#T57" title="AFF_4:th.57">AFF_4:57</a></i>;<br/><b>hence </b><a NAME="E5:9_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font>
 <b>by </b><i><a class="txt" href="#E3:9_1_1_3"><i><font color="Green" title="E6">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:9_1_1"/><b>then </b>
<font color="Maroon" title="c5">R2</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E7:9_1_1"/><b>then </b><i><font color="Green" title="E6">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:9_1_1_4"/><i><font color="Green" title="E7">A9</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> )
 <b>and </b><br/><a NAME="E2:9_1_1_4"/><i><font color="Green" title="E8">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font></span><br/><b>consider </b><font color="Maroon" title="c8">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:9_1_1_4"/><i><font color="Green" title="E9">A11</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c8">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_4"><i><font color="Green" title="E7">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:9_1_1_4"/><i><font color="Green" title="E10">A12</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c9">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_4"><i><font color="Green" title="E7">A9</font></i></a></i>;<br/><a NAME="E7:9_1_1_4"/>
<font color="Maroon" title="c8">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c9">Y</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:9_1_1_4"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="#E4:9_1_1_4"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="#E6:9_1_1_4"><i><font color="Green" title="E10">A12</font></i></a></i>;<br/><a NAME="E8:9_1_1_4"/><b>then </b>
<font color="Maroon" title="c9">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_4"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="#E6:9_1_1_4"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="aff_4.html#T58" title="AFF_4:th.58">AFF_4:58</a></i>;<br/><b>hence </b><a NAME="E9:9_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_4"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="#E6:9_1_1_4"><i><font color="Green" title="E10">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:9_1_1"/><b>then </b><i><font color="Green" title="E7">A13</font></i>: 
<font color="Maroon" title="c5">R2</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:9_1_1_5"/><i><font color="Green" title="E8">A14</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font> )
 <b>and </b><br/><a NAME="E2:9_1_1_5"/><i><font color="Green" title="E9">A15</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font></span><br/><b>consider </b><font color="Maroon" title="c9">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:9_1_1_5"/><i><font color="Green" title="E10">A16</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X</font> &amp; <font color="Maroon" title="c9">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_5"><i><font color="Green" title="E8">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:9_1_1_5"/><i><font color="Green" title="E11">A17</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c10">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_5"><i><font color="Green" title="E8">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:9_1_1_5"/><i><font color="Green" title="E12">A18</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Z</font> &amp; <font color="Maroon" title="c11">Z</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_5"><i><font color="Green" title="E8">A14</font></i></a></i>;<br/><a NAME="E9:9_1_1_5"/>
( <font color="Maroon" title="c9">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c10">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c11">Z</font> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:9_1_1_5"><i><font color="Green" title="E9">A15</font></i></a>, <a class="txt" href="#E4:9_1_1_5"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="#E6:9_1_1_5"><i><font color="Green" title="E11">A17</font></i></a>, <a class="txt" href="#E8:9_1_1_5"><i><font color="Green" title="E12">A18</font></i></a></i>;<br/><a NAME="E10:9_1_1_5"/><b>then </b>
<font color="Maroon" title="c9">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c11">Z</font>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_5"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="#E6:9_1_1_5"><i><font color="Green" title="E11">A17</font></i></a>, <a class="txt" href="#E8:9_1_1_5"><i><font color="Green" title="E12">A18</font></i></a>, <a class="ref" href="aff_4.html#T61" title="AFF_4:th.61">AFF_4:61</a></i>;<br/><b>hence </b><a NAME="E11:9_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_5"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="#E8:9_1_1_5"><i><font color="Green" title="E12">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:9_1_1"/><b>then </b>
<font color="Maroon" title="c5">R2</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:9_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> : ( <font color="Olive" title="b1">X</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  <a href="afproj.html#K2" title="AFPROJ:func.2">AfPlanes</a> <font color="Maroon" title="c1">AS</font>
 <b>by </b><i><a class="txt" href="#E7:9_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="#E9:9_1_1"><i><font color="Green" title="E7">A13</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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