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<a NAME="E1:22"/>
1,2 <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D2" title="ARYTM_3:def.2">ARYTM_3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c4">d1</font> or  not 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c5">d2</font> or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1"/><i><font color="Green" title="E20">A1</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c4">d1</font>
 <b>and </b><br/><a NAME="E2:22_1"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c5">d2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<b>thus </b><a NAME="E3:22_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="numbers.html#E1:22_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="ordinal3.html#T41" title="ORDINAL3:th.41">ORDINAL3:41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T39" title="ARYTM_3:th.39">ARYTM_3:39</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E21">A3</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T38" title="ARYTM_3:th.38">ARYTM_3:38</a></i>;<br/>
<a NAME="E4:22"/>
not 1 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> </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/>
<a NAME="E5:22"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</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="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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="numbers.html#E3:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:22"/><b>then </b>
 <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 
<b>by </b><i><a class="txt" href="numbers.html#E2:22"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="numbers.html#E8"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:22"/>
 <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <b>by </b><i><a class="txt" href="numbers.html#E26"><i><font color="Green" title="E18">Lm15</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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