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

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

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR5" target="_self">satisfies_minor_SCH*</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><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>, <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>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="aff_1.html#V1">being_line</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="aff_1.html#V1">being_line</a> )
 <b>by </b><i><a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E13</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>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><a NAME="E5:15_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:15_1_1"/><i><font color="Green">E14</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>13</sub></font> &amp; <font color="Maroon">c<sub>13</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="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E5:15_1_1"/>
not <font color="Maroon">c<sub>2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">c<sub>2</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>
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:15_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:15_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:15_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>3</sub></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="E5:15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:15_1_1"/><i><font color="Green">E16</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>12</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#T74">AFF_1:74</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<a NAME="E8:15_1_1"/>
<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>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E7:15_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E9:15_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:15_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E10:15_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:15_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E8:15_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:15_1_2"/><i><font color="Green">E15</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>14</sub></font> &amp; <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="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:15_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>13</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="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E5:15_1_2"/>
not <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:15_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:15_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:15_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E4:15_1_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>4</sub></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="E5:15_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</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 <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:15_1_2"/><i><font color="Green">E17</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp;  <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>16</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<a NAME="E8:15_1_2"/>
<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>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E7:15_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E9:15_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:15_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E10:15_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:15_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E11:15_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:15_1_3"/><i><font color="Green">E16</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>15</sub></font> &amp; <font color="Maroon">c<sub>15</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="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:15_1_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/>
<a NAME="E5:15_1_3"/>
not <font color="Maroon">c<sub>2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_3_1"/>
<font color="Maroon">c<sub>2</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>
 ;<br/>

<a NAME="E2:15_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E4:15_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E3:15_1_3_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>5</sub></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="E4:15_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:15_1_3"/><i><font color="Green">E18</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T74">AFF_1:74</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<a NAME="E8:15_1_3"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="conmetr1.html#E7:15_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:15_1_3"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="conmetr1.html#E2:15_1_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conmetr1.html#E4:15_1_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:15_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 ;<br/><a NAME="E14:15_1"/><i><font color="Green">E17</font></i>: 
( not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>15</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="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E4:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conmetr1.html#E10:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conmetr1.html#E13:15_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E15:15_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15"><i><font color="Green">E10</font></i></a>, <a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E4:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conmetr1.html#E10:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conmetr1.html#E13:15_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="conmetr1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E16:15_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E17:15_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E10:15_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E18:15_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E13:15_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E19:15_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E1:15"><i><font color="Green">E10</font></i></a>, <a class="txt" href="conmetr1.html#E1:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="conmetr1.html#E4:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="conmetr1.html#E7:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conmetr1.html#E10:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conmetr1.html#E13:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conmetr1.html#E14:15_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="conmetr1.html#E16:15_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="conmetr1.html#E17:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="conmetr1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E20:15_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="conmetr1.html#E13:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conmetr1.html#E14:15_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="conmetr1.html#E15:15_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
<font color="Maroon">c<sub>1</sub></font> <a href="conmetr1.html#NR1" target="_self">satisfies_minor_SCH</a> 
 <b>by </b><i><a class="ref" href="conmetr1.html#D1" target="_self">Def1</a></i>;<br/>


</div>
