<?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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E12</font></i>: 
not  <a href="analmetr.html#R5">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</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>4</sub></font> )
 
<b>by </b><i><a class="ref" href="euclmetr.html#T1" target="_self">Th1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:12"/><i><font color="Green">E14</font></i>: 
(  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp;  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E2:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="analmetr.html#D13">ANALMETR:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="conmetr.html#T3">CONMETR:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:12"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E3:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="conmetr.html#T3">CONMETR:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <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> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/>
<a NAME="E9:12"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="analmetr.html#NR6" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:12"><i><font color="Green">E15</font></i></a>, <a class="txt" href="euclmetr.html#E7:12"><i><font color="Green">E16</font></i></a>, <a class="ref" href="analmetr.html#T62">ANALMETR:62</a></i>;<br/>
<a NAME="E10:12"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="analmetr.html#T58">ANALMETR:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>12</sub></font> =  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> as   <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> <b>by </b><i><a class="ref" href="analmetr.html#T57">ANALMETR:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>4</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/>
<a NAME="E13:12"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T56">ANALMETR:56</a></i>;<br/>
<a NAME="E14:12"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>
<a NAME="E15:12"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="analmetr.html#R9">//</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b>
 <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#R10">_|_</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E5:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="analmetr.html#T73">ANALMETR:73</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b>
 <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analmetr.html#R9">//</a>  <a href="analmetr.html#K4">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E7:12"><i><font color="Green">E16</font></i></a>, <a class="ref" href="analmetr.html#T87">ANALMETR:87</a></i>;<br/>
<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64">ANALMETR:64</a></i>;<br/>
<a NAME="E6:12_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E14:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<a NAME="E7:12_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>15</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="euclmetr.html#E13:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E8:12_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="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E9:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclmetr.html#E1:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="analmetr.html#T55">ANALMETR:55</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <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> such that </b><br/><a NAME="E17:12"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E10:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</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/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E19:12"/>
( <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analmetr.html#R4">_|_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="euclmetr.html#E5:12"><i><font color="Green">E15</font></i></a>, <a class="txt" href="euclmetr.html#E7:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="euclmetr.html#E14:12"><i><font color="Green">E19</font></i></a>, <a class="txt" href="euclmetr.html#E17:12"><i><font color="Green">E20</font></i></a>, <a class="ref" href="analmetr.html#T78">ANALMETR:78</a></i>;<br/>


</div>
