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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) implies  ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:6"/><i><font color="Green" title="E9">A3</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]</span><br/><div class="add"><b>take </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]</span><br/><b>take </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c1">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> ) ) )</span><br/><b>thus </b><a NAME="E2:6_1_1_1"/>
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="arytm_3.html#E1:6_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> )</span><br/><b>thus </b><a NAME="E3:6_1_1_1"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:6"><i><font color="Green" title="E8">A2</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]</span><br/><div class="add"><b>take </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]</span><br/><b>take </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> ) ) )</span><br/><b>thus </b><a NAME="E2:6_1_1_2"/>
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="arytm_3.html#E1:6_1_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> )</span><br/><b>thus </b><a NAME="E3:6_1_1_2"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:6"><i><font color="Green" title="E8">A2</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:6"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green" title="E11">A7</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="arytm_3.html#E3:6"><i><font color="Green" title="E9">A3</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:6"/><i><font color="Green" title="E12">A8</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d1</font>, <font color="Olive" title="b3">d2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <br/>(  not <font color="Olive" title="b2">d1</font>,<font color="Olive" title="b3">d2</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  or  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b2">d1</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Olive" title="b3">d2</font> ) ) )
 <b>by </b><i><a class="txt" href="arytm_3.html#E5:6"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="arytm_3.html#E8:6"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E10:6"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>;<br/>
<a NAME="E11:6"/><b>then </b>
not <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="arytm_3.html#E8:6"><i><font color="Green" title="E12">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d1</font>, <font color="Maroon" title="c9">d2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E13:6"/><i><font color="Green" title="E13">A9</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c8">d1</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c9">d2</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="arytm_3.html#D2" target="_self" title="ARYTM_3:def.2">Def2</a></i>;<br/>
<a NAME="E14:6"/><i><font color="Green" title="E14">A10</font></i>: 
( <font color="Maroon" title="c8">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E8:6"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></i>;<br/>
<a NAME="E15:6"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c8">d1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c9">d2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal3.html#T39" title="ORDINAL3:th.39">ORDINAL3:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c11">d1</font> = <font color="Maroon" title="c8">d1</font>, <font color="Maroon" title="c12">d2</font> = <font color="Maroon" title="c9">d2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E17:6"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d1</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">d2</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c10">c</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a></i>;<br/>
<a NAME="E18:6"/><b>then </b><i><font color="Green" title="E15">A11</font></i>: 
<font color="Maroon" title="c12">d2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">d1</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E8:6"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="arytm_3.html#E14:6"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="ordinal3.html#T38" title="ORDINAL3:th.38">ORDINAL3:38</a></i>;<br/>
<div><i><font color="Green" title="E16">B13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">c'</font>, <font color="Maroon" title="c14">d1'</font>, <font color="Maroon" title="c15">d2'</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">d1'</font>,<font color="Maroon" title="c15">d2'</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c'</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d1'</font> implies  not <font color="Maroon" title="c12">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c'</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d2'</font> )</span><br/><b>assume </b><a NAME="E1:6_2"/><i><font color="Green" title="E17">A12</font></i>: 
( <font color="Maroon" title="c14">d1'</font>,<font color="Maroon" title="c15">d2'</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c'</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d1'</font> &amp; <font color="Maroon" title="c12">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c'</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d2'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:6_2"/><b>then </b>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c13">c'</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d1'</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c13">c'</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d2'</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>;<br/><b>hence </b><a NAME="E3:6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E8:6"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="arytm_3.html#E1:6_2"><i><font color="Green" title="E17">A12</font></i></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="E20:6"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c</font> or <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>
<a NAME="E21:6"/><b>then </b>
1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E13:6"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="arytm_3.html#E14:6"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="ordinal3.html#T17" title="ORDINAL3:th.17">ORDINAL3:17</a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></i>;<br/>
<a NAME="E22:6"/><b>then </b>
<font color="Maroon" title="c11">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></i>;<br/>
<b>hence </b><a NAME="E23:6"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E19:6"><i><font color="Green" title="E16">B13</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a>, <a class="txt" href="arytm_3.html#E6:6"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="arytm_3.html#E14:6"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="arytm_3.html#E18:6"><i><font color="Green" title="E15">A11</font></i></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>
