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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_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="E1:21_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c1">k</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="c2">n</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <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">b<sub>1</sub></font> being   <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>  st <br/>( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive" title="b2">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">w</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">w</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="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">w</font> &amp; ( for <font color="Olive" title="b1">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">w</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:21_1_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T14" target="_self" title="ARYTM_3:th.14">Th14</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">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">w</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">m</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">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c3">w</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font> )</span><br/><span class="kw">assume </span><a NAME="E4:21_1_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</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="c3">w</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font></span><br/><span class="kw">hence </span><a NAME="E5:21_1_1_1_1"/>
<font color="Maroon" title="c3">w</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_1"><span class="lab"><font color="Green" title="E15">A1</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="21_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( <font color="Maroon" title="c1">k</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="c2">n</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>  )
 ; <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">b<sub>1</sub></font> being   <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>  st <br/>( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive" title="b2">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b2">m</font> ) )</span><br/><div class="add"><a NAME="E2:21_1_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">k</font>
 ;<br/><a NAME="E3:21_1_1_1_2"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_2"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E4:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E5:21_1_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E6:21_1_1_1_2"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_2"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E7:21_1_1_1_2"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E8:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_3.html#E22"><span class="lab"><font color="Green" title="E14">Lm3</font></span></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">m</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/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">m</font> );<br/><a NAME="E9:21_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</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="E10:21_1_1_1_2"/>
( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></span>;<br/><a NAME="E11:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A6</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>2</sub></font>[<font color="Olive" title="b1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1_1_2"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E9:21_1_1_1_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="ordinal2.html#T58" title="ORDINAL2:th.58">ORDINAL2:58</a></span>;<br/><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="E13:21_1_1_1_2"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">A</font>]
 <span class="kw">and </span><br/><a NAME="E14:21_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</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">B</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="#E11:21_1_1_1_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c4">l</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E16:21_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font>
 <span class="kw">and </span><br/><a NAME="E17:21_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">l</font>
 <span class="kw">and </span><br/><a NAME="E18:21_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c4">l</font>
 <span class="kw">and </span><br/><a NAME="E19:21_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1_1_1_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">l</font> = <font color="Maroon" title="c4">l</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">l</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="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> &amp; ( for <font color="Olive" title="b1">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">l</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> ) )</span><br/><span class="kw">thus </span><a NAME="E21:21_1_1_1_2"/>
( <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21_1_1_1_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E18:21_1_1_1_2"><span class="lab"><font color="Green" title="E24">A11</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"> for <font color="Olive" title="b1">m</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 <font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">l</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">m</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">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font> implies <font color="Maroon" title="c5">l</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E22:21_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E23:21_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</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="c5">l</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font></span><br/><a NAME="E24:21_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></span>;<br/><a NAME="E25:21_1_1_1_2"/>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21_1_1_1_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></span>;<br/><a NAME="E26:21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>;<br/><a NAME="E27:21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></span>;<br/><a NAME="E28:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_1_1_1_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E24:21_1_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="arytm_3.html#T16" target="_self" title="ARYTM_3:th.16">Th16</a></span>;<br/><a NAME="E29:21_1_1_1_2"/>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:21_1_1_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></span>;<br/><a NAME="E30:21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></span>;<br/><a NAME="E31:21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></span>;<br/><a NAME="E32:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21_1_1_1_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E24:21_1_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="arytm_3.html#T16" target="_self" title="ARYTM_3:th.16">Th16</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 ;<br/><span class="kw">assume </span><a NAME="E2:21_1_1_1_2_1"/>
<font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</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> 
 ; <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="E3:21_1_1_1_2_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E4:21_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_1_1_1_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="arytm_3.html#T15" target="_self" title="ARYTM_3:th.15">Th15</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E5:21_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1_1_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E16:21_1_1_1_2"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E28:21_1_1_1_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E32:21_1_1_1_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21_1_1_1_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E4:21_1_1_1_2"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="arytm_3.html#T10" target="_self" title="ARYTM_3:th.10">Th10</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><span class="kw">end;</span></div><a NAME="E34:21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c5">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21_1_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>;<br/><span class="kw">hence </span><a NAME="E35:21_1_1_1_2"/>
<font color="Maroon" title="c5">l</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</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>
