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

<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>; <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> 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>  &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> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/> for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  holds <br/>(  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> or  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Y</font> implies  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E2:78"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E3:78"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:78"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Y</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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:78"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E7:78"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E8:78"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E10:78"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E11:78"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E12:78"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="aff_4.html#E47"><span class="lab"><font color="Green" title="E40">Lm9</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>);<br/>
<a NAME="E13:78"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E14:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A11</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/>
<a NAME="E15:78"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<a NAME="E16:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E17:78"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:78_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:78_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:78_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</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>
<a NAME="E18:78"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:78"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E19:78"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>) &amp; <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E20:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A17</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="aff_4.html#E42"><span class="lab"><font color="Green" title="E35">Lm8</font></span></a></span>;<br/>
<a NAME="E21:78"/><span class="lab"><font color="Green" title="E81">A18</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E15:78"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></span>;<br/>
<a NAME="E22:78"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E23:78"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="aff_4.html#E42"><span class="lab"><font color="Green" title="E35">Lm8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:78"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="txt" href="#E18:78"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E21:78"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="txt" href="#E20:78"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E17:78"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="ref" href="aff_4.html#T26" target="_self" title="AFF_4:th.26">Th26</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>
