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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_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="16_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
( <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:()"><i><font color="Red">::  thesis: </font></i></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"><b>reconsider </b><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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">w</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><a NAME="E3:16_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> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T14" target="_self" title="ARYTM_3:th.14">Th14</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">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E4:16_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:()"><i><font color="Red">::  thesis: </font></i></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/><b>hence </b><a NAME="E5:16_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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:16_1_1_1_1"><i><font color="Green" title="E16">A1</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="16_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2"/><i><font color="Green" title="E16">A2</font></i>: 
( <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:16_1_1_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
( <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> )
 <b>by </b><i><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> 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="E3:16_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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:16_1_1_1_2"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:16_1_1_1_2"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:16_1_1_1_2"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E6:16_1_1_1_2"/><b>then </b>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E22"><i><font color="Green" title="E15">Lm3</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/><a NAME="E7:16_1_1_1_2"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c1">k</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T58" title="ORDINAL2:th.58">ORDINAL2:58</a></i>;<br/><a NAME="E8:16_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A4</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>2</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:16_1_1_1_2"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><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="E10:16_1_1_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">A</font>]
 <b>and </b><br/><a NAME="E11:16_1_1_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
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>
 <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#E8:16_1_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>);<br/></i><b>consider </b><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><b> such that </b><br/><a NAME="E13:16_1_1_1_2"/><i><font color="Green" title="E21">A7</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <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> &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">l</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E10:16_1_1_1_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">l</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><a NAME="E15:16_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> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:16_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></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">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E16:16_1_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:16_1_1_1_2_1"/>
 <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E2:16_1_1_1_2_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:16_1_1_1_2"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:16_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:16_1_1_1_2_1"/><i><font color="Green" title="E24">A10</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="ordinal3.html#T78" title="ORDINAL3:th.78">ORDINAL3:78</a></i>;<br/><a NAME="E5:16_1_1_1_2_1"/>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:16_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</a></i>;<br/><a NAME="E6:16_1_1_1_2_1"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>;<br/><a NAME="E7:16_1_1_1_2_1"/><b>then </b>
( <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> &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> <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> )
 <b>by </b><i><a class="ref" href="arytm_3.html#D3" target="_self" title="ARYTM_3:def.3">Def3</a></i>;<br/><a NAME="E8:16_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
( <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> &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> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c5">l</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E16:16_1_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="arytm_3.html#E4:16_1_1_1_2_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="arytm_3.html#T16" target="_self" title="ARYTM_3:th.16">Th16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1_1_2_1_1"/><i><font color="Green" title="E26">A12</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:16_1_1_1_2_1_1"/>
 <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:16_1_1_1_2_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:16_1_1_1_2_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:16_1_1_1_2_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E8:16_1_1_1_2_1"><i><font color="Green" title="E25">A11</font></i></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></i>;<br/><a NAME="E5:16_1_1_1_2_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E11:16_1_1_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="arytm_3.html#E13:16_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="arytm_3.html#E8:16_1_1_1_2_1"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:16_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E13:16_1_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="arytm_3.html#E3:16_1_1_1_2_1"><i><font color="Green" title="E23">A9</font></i></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></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="E10:16_1_1_1_2_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:16_1_1_1_2_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>;<br/><b>hence </b><a NAME="E11:16_1_1_1_2_1"/>
<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>
 <b>by </b><i><a class="ref" href="arytm_3.html#T11" target="_self" title="ARYTM_3:th.11">Th11</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><b>hence </b><a NAME="E18:16_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>
 ; <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>
