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<div class="add">

<b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/>

<b>set </b><font color="Maroon">AS</font> =  <a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font>;<br/>
<a NAME="E1:34"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">d</font> &amp; <font color="Olive">a</font>,<font color="Olive">d</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> holds <br/><font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:34_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> )
 <b>and </b><br/><a NAME="E2:34_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 ;<br/>

<b>assume </b><a NAME="E3:34_1"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E4:34_1"/><b>then </b><i><font color="Green">E43</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font>, <font color="Maroon">d1</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> <b>by </b><i/>;<br/>
<a NAME="E6:34_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c1</font>,<font color="Maroon">d1</font> &amp; <font color="Maroon">a1</font>,<font color="Maroon">c1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">b1</font>,<font color="Maroon">d1</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:34_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<a NAME="E7:34_1"/>
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1"/>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">c1</font>
 ;<br/>

<a NAME="E2:34_1_1"/><b>then </b>
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">a1</font>,<font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="papdesaf.html#T4" target="_self">Th4</a>, <a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E9:34_1"/><i><font color="Green">E60</font></i>: 
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x1</font>,<font color="Maroon">a1</font>,<font color="Maroon">d1</font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">x1</font>,<font color="Maroon">b1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="papdesaf.html#T5" target="_self">Th5</a>, <a class="ref" href="pasch.html#T33">PASCH:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E11:34_1"/><i><font color="Green">E61</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">d</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_2"/>
( <font color="Maroon">x1</font>,<font color="Maroon">a1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x1</font>,<font color="Maroon">d1</font> &amp; <font color="Maroon">x1</font>,<font color="Maroon">b1</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">x1</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E2:34_1_2"/><b>then </b>
( <font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/>
<a NAME="E3:34_1_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>,<font color="Maroon">d</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">d</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">P</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font>;<br/>
<b>set </b><font color="Maroon">Q</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E12:34_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E13:34_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:34_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E14:34_1"/><b>then </b><i><font color="Green">E65</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E15:34_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E16:34_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E17:34_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:34_1"><i><font color="Green">E29</font></i></a>, , , , <a class="txt" href="papdesaf.html#E3:34_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T52">AFF_1:52</a></i>;<br/>
<a NAME="E18:34_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:34_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E19:34_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="papdesaf.html#E3:34_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34"/>
 <a href="diraf.html#K2">Lambda</a> <font color="Maroon">OAS</font> <a href="papdesaf.html#NR5" target="_self">satisfies_Fano</a> 
 <b>by </b><i/>;<br/>


</div>
