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<div class="add">

<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/>

<b>thus </b><a NAME="E1:1"/>
( <font color="Maroon">AFP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a>  implies for <font color="Olive">o</font>, <font color="Olive">a</font>, <font color="Olive">a'</font>, <font color="Olive">p</font>, <font color="Olive">p'</font>, <font color="Olive">q</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">p</font>,<font color="Olive">p'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">q</font>,<font color="Olive">q'</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">p</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">p'</font> &amp; <font color="Olive">a</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">q'</font> holds <br/><font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">AFP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 ;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>













<b>assume </b><a NAME="E2:1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">p'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">q'</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">p'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">q'</font> )
 ;<br/>

<a NAME="E3:1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>;<br/>
<a NAME="E4:1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</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="E5:1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E6:1_1"/><b>then </b><i><font color="Green">E48</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">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E7:1_1"/><i><font color="Green">E49</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<a NAME="E8:1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:1_1"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">o</font>, <font color="Olive">a</font>, <font color="Olive">a'</font>, <font color="Olive">p</font>, <font color="Olive">p'</font>, <font color="Olive">q</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  st  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">p</font>,<font color="Olive">p'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">q</font>,<font color="Olive">q'</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">p</font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">o</font>,<font color="Olive">a</font>,<font color="Olive">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">p'</font> &amp; <font color="Olive">a</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">q'</font> holds <br/><font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Olive">q'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">C</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a'</font>,<font color="Maroon">c'</font> )
 ;<br/><a NAME="E2:1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>,<font color="Maroon">b'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E3:1_2"/><i><font color="Green">E53</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:1_2"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:1_2_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:1_2"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1"/>
<font color="Maroon">AFP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 <b>by </b><i><a class="ref" href="aff_2.html#D4">AFF_2:def 4</a></i>;<br/>


</div>
