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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="analmetr.html#NM2">OrtAfPl</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E21</font></i>: 
 <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:21_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>and </b><br/><a NAME="E2:21_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E3:21_1_1"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:21_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E5:21_1_1"/><i><font color="Green">E26</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E6:21_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E7:21_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><a NAME="E9:21_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><a NAME="E10:21_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E11:21_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E3:21_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="euclmetr.html#E4:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euclmetr.html#E7:21_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/><a NAME="E12:21_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="analmetr.html#T81">ANALMETR:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:21_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:21_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E4:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euclmetr.html#E7:21_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T41">AFF_1:41</a></i>;<br/>
<a NAME="E2:21_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:21_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E15:21_1_1"/><i><font color="Green">E35</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E3:21_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="euclmetr.html#E4:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:21_1_1"/><i><font color="Green">E36</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_4"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_4"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E3:21_1_1_4"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E2:21_1_1_4"><i><font color="Green">E37</font></i></a>, <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="E4:21_1_1_4"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euclmetr.html#E7:21_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#D5">AFF_1:def 5</a></i>;<br/>
<a NAME="E5:21_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E3:21_1_1_4"><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="E6:21_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euclmetr.html#E15:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:21_1_1_5"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclmetr.html#E16:21_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/><a NAME="E3:21_1_1_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclmetr.html#E9:21_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euclmetr.html#E15:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euclmetr.html#E1:21_1_1_5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_5"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_1_1_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T12">AFF_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_6"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_6_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:21_1_1_6_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E3:21_1_1_6_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclmetr.html#E16:21_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/><a NAME="E4:21_1_1_6_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclmetr.html#E9:21_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euclmetr.html#E15:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1_6_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/><b>hence </b><a NAME="E5:21_1_1_6_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:21_1_1_6_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_6_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:21_1_1_6_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_1_1_6_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1_1_6"><i><font color="Green">E38</font></i></a></i>;<br/>

<a NAME="E2:21_1_1_6_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E6:21_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:21_1_1_6_2_1"/>
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E15:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euclmetr.html#E16:21_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_6_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E9:21_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euclmetr.html#E1:21_1_1_6"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1_6_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="aff_1.html#T69">AFF_1:69</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:21_1_1_6_2"/><i><font color="Green">E42</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_6_2_2"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_6_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:21_1_1_6_2_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_6_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E15:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euclmetr.html#E1:21_1_1_6_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1_6_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:21_1_1_6_2"/>
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_6_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_6_2_3"/>
 <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_6_2_3"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>
<a NAME="E3:21_1_1_6_2_3"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_6_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E16:21_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euclmetr.html#E1:21_1_1_6_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1_6_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:21_1_1_6_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclmetr.html#E10:21_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euclmetr.html#E11:21_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euclmetr.html#E12:21_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="euclmetr.html#E13:21_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="euclmetr.html#E14:21_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euclmetr.html#E2:21_1_1_6_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="euclmetr.html#E3:21_1_1_6_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="conmetr.html#D5">CONMETR:def 5</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_6_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_1_6"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_1_1_6"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:21_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E17:21_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:21_1"/><b>then </b>
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> is <a href="aff_2.html#V7">Moufangian</a>
 <b>by </b><i><a class="ref" href="aff_2.html#D7">AFF_2:def 7</a></i>;<br/><b>hence </b><a NAME="E4:21_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="euclmetr.html#V5" target="_self">Moufangian</a>
 <b>by </b><i><a class="ref" href="euclmetr.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="euclmetr.html#V5" target="_self">Moufangian</a>
 ;<br/><a NAME="E2:21_2"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font> <a href="papdesaf.html#NR3" target="_self">satisfies_TDES'</a> 
 <b>by </b><i><a class="ref" href="euclmetr.html#D5" target="_self">Def5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:21_2_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:21_2_1"/><i><font color="Green">E23</font></i>: 
( not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E3:21_2_1"/><i><font color="Green">E24</font></i>: 
(  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E4:21_2_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3">Af</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T47">ANALMETR:47</a></i>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E6:21_2_1"/><i><font color="Green">E26</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E7:21_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:21_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E8:21_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E9:21_2_1"/><i><font color="Green">E28</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:21_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/><a NAME="E10:21_2_1"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:21_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E1:21_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E9:21_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>
<a NAME="E3:21_2_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E6:21_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E1:21_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:21_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:21_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:21_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:21_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<a NAME="E2:21_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E1:21_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="aff_1.html#D4">AFF_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R2">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:21_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:21_2_1_3"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E4:21_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/>
<b>hence </b><a NAME="E2:21_2_1_3"/>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:21_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E4:21_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/><a NAME="E14:21_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euclmetr.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E15:21_2_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="euclmetr.html#E2:21_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclmetr.html#E6:21_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euclmetr.html#E8:21_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclmetr.html#E9:21_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euclmetr.html#E10:21_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euclmetr.html#E11:21_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euclmetr.html#E12:21_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euclmetr.html#E13:21_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_2.html#D7">AFF_2:def 7</a></i>;<br/><b>hence </b><a NAME="E16:21_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48">ANALMETR:48</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:21_2"/>
 <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="conmetr.html#D5">CONMETR:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="euclmetr.html#V5" target="_self">Moufangian</a> iff  <a href="conmetr.html#NR5" target="_self">TDES_holds_in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:21"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
