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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#K2">OASpace</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:25"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>7</sub></font> &amp; not  <a href="diraf.html#R3">LIN</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; not  <a href="diraf.html#R3">LIN</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </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>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1"/><i><font color="Green">E20</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>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</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>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:25_1"/><i><font color="Green">E21</font></i>: 
( not  <a href="diraf.html#R3">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; not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green">E22</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>8</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>9</sub></font>,<font color="Maroon">c<sub>7</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="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:25_1"/><i><font color="Green">E23</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>4</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>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:25_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green">E27</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>8</sub></font>
 ;<br/>

<a NAME="E2:25_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E3:25_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E4:25_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E5:25_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E6:25_1_1_1"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E7:25_1_1_1"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_2"/><i><font color="Green">E27</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>9</sub></font>
 ;<br/>

<a NAME="E2:25_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E3:25_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E4:25_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<a NAME="E5:25_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27">DIRAF:27</a></i>;<br/>
<a NAME="E6:25_1_1_2"/><b>then </b>
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>
<a NAME="E7:25_1_1_2"/><b>then </b>
(  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="diraf.html#R3">LIN</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="diraf.html#T38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1"/>
<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>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green">E25</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>7</sub></font>
 ;<br/><a NAME="E2:25_1_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="geomtrap.html#T2">GEOMTRAP:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:25_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:25_1_2"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E5:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="papdesaf.html#E1:25_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="papdesaf.html#E32"><i><font color="Green">Lemma17</font></i></a></i>;<br/><a NAME="E6:25_1_2"/>
( <font color="Maroon">c<sub>3</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>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E7:25_1_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="geomtrap.html#T2">GEOMTRAP:2</a></i>;<br/><a NAME="E8:25_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="geomtrap.html#R1">'||'</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>,<font color="Maroon">c<sub>13</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E9:25_1_2"/>
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="txt" href="papdesaf.html#E3:25_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="geomtrap.html#T2">GEOMTRAP:2</a></i>;<br/><a NAME="E10:25_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1">GEOMTRAP:def 1</a></i>;<br/><a NAME="E11:25_1_2"/>
( not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_2_1"/>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="geomtrap.html#R1">'||'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<a NAME="E2:25_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#D1">GEOMTRAP:def 1</a></i>;<br/>
<a NAME="E3:25_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="geomtrap.html#T2">GEOMTRAP:2</a></i>;<br/>
<a NAME="E4:25_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:25_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="diraf.html#D5">DIRAF:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:25_1_2"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:25_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="papdesaf.html#E5:25_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="papdesaf.html#E8:25_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="papdesaf.html#E10:25_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="papdesaf.html#T15" target="_self">Th15</a></i>;<br/><b>then </b><i><font color="Green">E30</font></i>: 1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224">XCMPLX_1:224</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
;<br/><a NAME="E14:25_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K2">"</a>  )
 <b>by </b><i><a class="txt" href="papdesaf.html#E5:25_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E15:25_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="analoaf.html#R1">//</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="papdesaf.html#E13:25_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="analoaf.html#D1">ANALOAF:def 1</a></i>;<br/><a NAME="E16:25_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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="geomtrap.html#T2">GEOMTRAP:2</a></i>;<br/><b>hence </b><a NAME="E17:25_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>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:25_1"/>
<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>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E6:25_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon">c<sub>2</sub></font> <a href="papdesaf.html#NR7" target="_self">satisfies_DES_1</a> 
 <b>by </b><i><a class="ref" href="papdesaf.html#D16" target="_self">Def16</a></i>;<br/>


</div>
