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

<span class="kw">set </span><font color="Maroon" title="c2">X</font> =  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">XXX</font> = <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">R1</font>, <font color="Maroon" title="c5">R2</font> be    <a href="collsp.html#M1" title="COLLSP:mode.1">Relation3</a> of  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b6">r</font> &amp;  not <font color="Olive" title="b4">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b5">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b6">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  ) ) ) &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b6">r</font> &amp;  not <font color="Olive" title="b4">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b5">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b6">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  ) ) ) implies <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1_2"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b6">r</font> &amp;  not <font color="Olive" title="b4">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b5">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b6">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  ) )
 <span class="kw">and </span><br/><a NAME="E2:39_1_2"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b6">r</font> &amp;  not <font color="Olive" title="b4">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b5">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b6">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font></span><br/>

<a NAME="E3:39_1_2"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c5">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D1" title="COLLSP:def.1">COLLSP:def 1</a></span>;<br/>
<a NAME="E4:39_1_2"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c4">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D1" title="COLLSP:def.1">COLLSP:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> iff <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )</span><br/><div><span class="lab"><font color="Green" title="E33">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<font color="Maroon" title="c6">u</font> <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:39_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_2"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="domain_1.html#T3" title="DOMAIN_1:th.3">DOMAIN_1:3</a></span>;<br/><a NAME="E4:39_1_2_1_1"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b3">r</font> &amp;  not <font color="Olive" title="b1">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b2">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b3">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E1:39_1_2_1_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E3:39_1_2_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:39_1_2_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E3:39_1_2_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_2_1_2"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">ProjectivePoints</a> <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:39_1_2_1_2"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_2"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="domain_1.html#T3" title="DOMAIN_1:th.3">DOMAIN_1:3</a></span>;<br/><a NAME="E4:39_1_2_1_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Olive" title="b3">r</font> &amp;  not <font color="Olive" title="b1">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b2">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Olive" title="b3">r</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E1:39_1_2_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E3:39_1_2_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:39_1_2_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E3:39_1_2_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:39_1_2_1"/>
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R1</font> iff <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_2_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:39_1_2"/>
<font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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