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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:10"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 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="10_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:10_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="10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">take </span><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">take </span><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c1">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E2:10_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:10_1_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E3:10_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_2"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">take </span><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">take </span><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E2:10_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:10_1_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E3:10_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E5:10"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]
 <span class="kw">and </span><br/><a NAME="E6:10"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
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>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E8">A3</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E8:10"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:10"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E10:10"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E11:10"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E13:10"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>;<br/>
<a NAME="E14:10"/><span class="kw">then </span>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A11</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E16:10"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E17:10"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E18:10"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D2" target="_self" title="ARYTM_3:def.2">Def2</a></span>;<br/>
<a NAME="E19:10"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></span>;<br/>
<a NAME="E20:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="ordinal3.html#T39" title="ORDINAL3:th.39">ORDINAL3:39</a></span>;<br/>
<a NAME="E21:10"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></span>;<br/>
<a NAME="E22:10"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="ordinal3.html#T39" title="ORDINAL3:th.39">ORDINAL3:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:10"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E24:10"/>
( 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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:10"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/>
<a NAME="E25:10"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E19:10"><span class="lab"><font color="Green" title="E18">A15</font></span></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></span>;<br/>
<a NAME="E26:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">c9</font>, <font color="Maroon" title="c14">d19</font>, <font color="Maroon" title="c15">d29</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">d19</font>,<font color="Maroon" title="c15">d29</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">c9</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d19</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">c9</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d29</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_2"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
<font color="Maroon" title="c14">d19</font>,<font color="Maroon" title="c15">d29</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:10_2"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
( <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c9</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d19</font> &amp; <font color="Maroon" title="c12">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c9</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:10_2"/>
( <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">c9</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c14">d19</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">c9</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c15">d29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E2:10_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>;<br/><span class="kw">hence </span><a NAME="E4:10_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E1:10_2"><span class="lab"><font color="Green" title="E23">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:10"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E16">A13</font></span></a></span>;<br/>
<a NAME="E29:10"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E19:10"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="ordinal3.html#T38" title="ORDINAL3:th.38">ORDINAL3:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E27:10"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
