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

<b>let </b><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="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:15"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="arytm_0.html#K2">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be  <a href="xreal_0.html#V1" target="_self">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E4:15_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/>
<a NAME="E6:15_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="xreal_0.html#E3:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E4:15_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E7:15_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="xreal_0.html#E3:15_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E4:15_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b>
(  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<b>thus </b> <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> = 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,0
<b>by </b><i><a class="txt" href="xreal_0.html#E1:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="xreal_0.html#E6:15_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="xreal_0.html#E7:15_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="xreal_0.html#E5:15_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="xreal_0.html#E9:15_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font> )
 
;<br/>
<a NAME="E6:15"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:15"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
<b>by </b><i><a class="txt" href="xreal_0.html#E2:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:15"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E9:15"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2"/>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="xreal_0.html#E1:15"><i><font color="Green">E22</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E2:15_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:15_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E5:15_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:15_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E1:15"><i><font color="Green">E22</font></i></a>, <a class="txt" href="xreal_0.html#E1:15_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="xreal_0.html#E2:15_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E8:15_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E9:15_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:15_2_1"/><i><font color="Green">E35</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E1:15_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E12:15_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E13:15_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E14:15_2_1"/><i><font color="Green">E38</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E2:15_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><a NAME="E15:15_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>18</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E4:15_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="xreal_0.html#E5:15_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="xreal_0.html#E6:15_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="xreal_0.html#E8:15_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="xreal_0.html#E12:15_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="arytm_1.html#T8">ARYTM_1:8</a></i>;<br/><b>hence </b><a NAME="E16:15_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_0.html#E5:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="xreal_0.html#E8:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="xreal_0.html#E9:15_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="xreal_0.html#E10:15_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="xreal_0.html#E13:15_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="xreal_0.html#E14:15_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="xreal_0.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:15_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:15_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:15_2_2"/><i><font color="Green">E30</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E9:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="xreal_0.html#E1:15_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E8:15_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E9:15_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:15_2_2"/><i><font color="Green">E31</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E2:15_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><a NAME="E11:15_2_2"/>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="xreal_0.html#E6:15_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="xreal_0.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:15_2_2"/><b>then </b>
( not  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; not  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_0.html#E10:15_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="arytm_0.html#T5">ARYTM_0:5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E13:15_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_0.html#E5:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="xreal_0.html#E8:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:15_2_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:15_2_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:15_2_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:15_2_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:15_2_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E1:15"><i><font color="Green">E22</font></i></a>, <a class="txt" href="xreal_0.html#E1:15_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="xreal_0.html#E2:15_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="xreal_0.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E8:15_2_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:15_2_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:15_2_3"/><i><font color="Green">E35</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E9:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="xreal_0.html#E1:15_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E12:15_2_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:15_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E14:15_2_3"/><i><font color="Green">E38</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xreal_0.html#E9:15"><i><font color="Green">E27</font></i></a>, <a class="txt" href="xreal_0.html#E2:15_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><a NAME="E15:15_2_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E4:15_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="xreal_0.html#E8:15_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E16:15_2_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E5:15_2_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="xreal_0.html#E12:15_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:15_2_3"/><i><font color="Green">E41</font></i>: 
(  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="xreal_0.html#E10:15_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="xreal_0.html#E14:15_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="xreal_0.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E18:15_2_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>16</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E6:15_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="xreal_0.html#E15:15_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="xreal_0.html#E16:15_2_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="arytm_1.html#T8">ARYTM_1:8</a></i>;<br/><b>hence </b><a NAME="E19:15_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_0.html#E5:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="xreal_0.html#E8:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="xreal_0.html#E9:15_2_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="xreal_0.html#E10:15_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="xreal_0.html#E13:15_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="xreal_0.html#E14:15_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="xreal_0.html#E17:15_2_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="xreal_0.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
