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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> = 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E1:10"/><i><font color="Green" title="E8">A1</font></i>: 
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/>
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green" title="E9">A2</font></i>: 
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:10_1"/>
<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:10_1"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><a NAME="E6:10_1"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3:10_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E7:10_1"/><i><font color="Green" title="E11">A4</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3:10_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div><i><font color="Green" title="E12">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font>
 ;<br/><a NAME="E2:10_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><a NAME="E3:10_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><a NAME="E4:10_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="numbers.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="numbers.html#E3:10_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E5:10_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:10_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_2"/>
( ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="numbers.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="numbers.html#E6:10_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="numbers.html#E7:10_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1"><b>suppose </b></a><a NAME="E1:10_1_2_1"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E8:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" title="ZFMISC_1:th.9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_2"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_2"/><i><font color="Green" title="E13">A6</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E2:10_1_2_2"/><i><font color="Green" title="E14">A7</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:10_1_2_2"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E2:10_1_2_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="numbers.html#E10"><i><font color="Green" title="E6">Lm6</font></i></a></i>;<br/><a NAME="E4:10_1_2_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E1:10_1_2_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E5:10_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:10_1_2_2"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:10_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E3:10_1_2_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_3"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2_3"/><i><font color="Green" title="E13">A9</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E2:10_1_2_3"/><i><font color="Green" title="E14">A10</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E3:10_1_2_3"/><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E2:10_1_2_3"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="numbers.html#E10"><i><font color="Green" title="E6">Lm6</font></i></a></i>;<br/><a NAME="E4:10_1_2_3"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="numbers.html#E1:10_1_2_3"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E5:10_1_2_3"/><b>then </b>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:10_1_2_3"/><b>then </b>
 <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:10_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E3:10_1_2_3"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_4"><b>suppose </b></a><a NAME="E1:10_1_2_4"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E8:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T9" title="ZFMISC_1:th.9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:10"/><b>then </b><i><font color="Green" title="E9">A12</font></i>: 
not 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  ) </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/>
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><a NAME="E2:10_2"/><b>then </b>
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> is <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E3:10_2"/><b>then </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/><b>hence </b><a NAME="E4:10_2"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E10">A13</font></i>: 
not 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E3:10"><i><font color="Green" title="E9">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">IR</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> ;<br/>
<a NAME="E6:10"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">A</font> where <font color="Olive" title="b3">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : for <font color="Olive" title="b4">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> holds <br/>( ( for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b5">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b4">r</font> holds <br/><font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> ) &amp;  ex <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b5">s</font> ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:10_3"/><i><font color="Green" title="E11">A14</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  : for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>( ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) &amp;  ex <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b3">s</font> ) ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E4:10_3"/><i><font color="Green" title="E12">A15</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E5:10_3"/><i><font color="Green" title="E13">A16</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/>( ( for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> ) &amp;  ex <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b2">s</font> ) )
 <b>by </b><i><a class="txt" href="numbers.html#E2:10_3"><i><font color="Green" title="E11">A14</font></i></a></i>;<br/>
<a NAME="E6:10_3"/><i><font color="Green" title="E14">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E4:10_3"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:10_3"/>
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E5:10_3"><i><font color="Green" title="E13">A16</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">r1</font>, <font color="Maroon" title="c9">r2</font>, <font color="Maroon" title="c10">r3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E9:10_3"/><i><font color="Green" title="E15">A18</font></i>: 
( <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c10">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> )
 <b>and </b><br/><a NAME="E10:10_3"/><i><font color="Green" title="E16">A19</font></i>: 
( <font color="Maroon" title="c8">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">r2</font> &amp; <font color="Maroon" title="c9">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r3</font> &amp; <font color="Maroon" title="c10">r3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">r1</font> )
 <b>by </b><i><a class="txt" href="numbers.html#E6:10_3"><i><font color="Green" title="E14">A17</font></i></a>, <a class="ref" href="arytm_3.html#T82" title="ARYTM_3:th.82">ARYTM_3:82</a></i>;<br/>
<a NAME="E11:10_3"/>
( ( <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c10">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c10">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 
<b>by </b><i><a class="txt" href="numbers.html#E4:10_3"><i><font color="Green" title="E12">A15</font></i></a>, <a class="txt" href="numbers.html#E9:10_3"><i><font color="Green" title="E15">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:10_3"/>
contradiction
 <b>by </b><i><a class="txt" href="numbers.html#E10:10_3"><i><font color="Green" title="E16">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10"/><b>then </b>
not 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K1" title="ARYTM_2:func.1">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="arytm_2.html#D1" title="ARYTM_2:def.1">ARYTM_2:def 1</a></i>;<br/>
<a NAME="E8:10"/><b>then </b><i><font color="Green" title="E11">A20</font></i>: 
not 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D2" title="ARYTM_2:def.2">ARYTM_2:def 2</a>, <a class="txt" href="numbers.html#E5:10"><i><font color="Green" title="E10">A13</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_4"/>
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_4"/><i><font color="Green" title="E12">A21</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E4:10_4"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <b>and </b><br/><a NAME="E5:10_4"/><i><font color="Green" title="E13">A22</font></i>: 
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><a NAME="E6:10_4"/><i><font color="Green" title="E14">A23</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="numbers.html#E3:10_4"><i><font color="Green" title="E12">A21</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:10_4"/>
0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,<font color="Maroon" title="c1">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></i>;<br/><a NAME="E8:10_4"/><b>then </b><i><font color="Green" title="E15">A24</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_4_1"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><b>by </b><i><a class="txt" href="numbers.html#E5:10_4"><i><font color="Green" title="E13">A22</font></i></a>, <a class="txt" href="numbers.html#E6:10_4"><i><font color="Green" title="E14">A23</font></i></a>, <a class="txt" href="numbers.html#E8:10_4"><i><font color="Green" title="E15">A24</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_4_1_1"><b>suppose </b></a><a NAME="E1:10_4_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:10_4_1_1"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:10_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_4_1_2"><b>suppose </b></a><a NAME="E1:10_4_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><div class="add"><a NAME="E2:10_4_1_2"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> ,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:10_4_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:10"/>
not 0,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a>  <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E8:10"><i><font color="Green" title="E11">A20</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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