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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:12"/><i><font color="Green">E17</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T71">FUNCT_4:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E18</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:12_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="arytm_3.html#R1">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:12_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E6:12_1"/><i><font color="Green">E19</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:12_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><a NAME="E7:12_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:12_1"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E6:12_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:12_1"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><a NAME="E10:12_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:12_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E3:12_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="arytm_0.html#E6:12_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E4:12_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="arytm_0.html#E3:12_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:12_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:12_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_2"/>
( ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="arytm_0.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="arytm_0.html#E7:12_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="arytm_0.html#E8:12_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_1"><b>suppose </b></a><a NAME="E1:12_1_2_1"/>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:12_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:12_1_2_2"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:12_1_2_2"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:12_1_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:12_1_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="arytm_0.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/><a NAME="E4:12_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12_1_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:12_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:12_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:12_1_2_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:12_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:12_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:12_1_2_3"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:12_1_2_3"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:12_1_2_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:12_1_2_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="arytm_0.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/><a NAME="E4:12_1_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:12_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E10:12_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:12_1_2_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:12_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:12_1_2_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_4"><b>suppose </b></a><a NAME="E1:12_1_2_4"/>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E11:12_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:12"/><b>then </b><i><font color="Green">E18</font></i>: 
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R1">are_relative_prime</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E2:12_2"/><b>then </b>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="ordinal1.html#V3">ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:12_2"/><b>then </b>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><b>hence </b><a NAME="E4:12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:12"/><b>then </b><i><font color="Green">E19</font></i>: 
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="txt" href="arytm_0.html#E3:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="arytm_3.html#D7">ARYTM_3:def 7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> ;<br/>
<a NAME="E6:12"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>5</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>5</sub></font> ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:12_3"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:12_3"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:12_3"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:12_3"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:12_3"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_0.html#E3:12_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:12_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_0.html#E4:12_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>then consider </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> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:12_3"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E9:12_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:12_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="arytm_3.html#T82">ARYTM_3:82</a></i>;<br/>
<a NAME="E10:12_3"/>
( ( <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>5</sub></font></span><a href="tarski.html#K4">]</a></span> or <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">1,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <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>5</sub></font></span><a href="tarski.html#K4">]</a></span> or <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">1,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( <font color="Maroon">c<sub>10</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>5</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> ) )
 
<b>by </b><i><a class="txt" href="arytm_0.html#E3:12_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="arytm_0.html#E8:12_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:12_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E9:12_3"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/><b>then </b>
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_0.html#E1:12"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E8:12"/><b>then </b>
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D1">ARYTM_2:def 1</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="xboole_0.html#K2">\/</a> <a href="arytm_2.html#K1">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="txt" href="arytm_0.html#E5:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:12"/><b>then </b><i><font color="Green">E20</font></i>: 
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D2">ARYTM_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_4"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</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"><a href="arytm_3.html#K13">{}</a> </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/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E4:12_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 <b>and </b><br/><a NAME="E5:12_4"/><i><font color="Green">E22</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><a NAME="E6:12_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="arytm_0.html#E3:12_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:12_4"/><b>then </b><i><font color="Green">E23</font></i>: 
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_0.html#E5:12_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><a NAME="E8:12_4"/>
0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T71">FUNCT_4:71</a></i>;<br/><a NAME="E9:12_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E10:12_4"/><b>then </b><i><font color="Green">E24</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E7:12_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_4_1"/>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> )
 </span><b>by </b><i><a class="txt" href="arytm_0.html#E10:12_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_1"><b>suppose </b></a><a NAME="E1:12_4_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:12_4_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_4_1_2"><b>suppose </b></a><a NAME="E1:12_4_1_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><a NAME="E2:12_4_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:12_4_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E12:12"/><b>then </b>
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</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"><a href="arytm_3.html#K13">{}</a> </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="arytm_0.html#E10:12"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:12"/>
not 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a></i>;<br/>


</div>
