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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> );<br/>
<a NAME="E1:48_1_1"/>
( 1 <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; 1 <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:48_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E2:48_1_1_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="nat_1.html#E3:48_1_1"><i><font color="Green">E32</font></i></a>, </i>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E3:48_1_1_1"/>
( <font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 ;<br/><b>hence </b><a NAME="E4:48_1_1_1"/>
<font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:48_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:48_1_1"><i><font color="Green">E32</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:48_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" target="_self">NAT_1:sch 6</a>(, );<br/></i><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E6:48_1_1_2"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E7:48_1_1_2"/>
( <font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E8:48_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:48_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E10:48_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self">Th2</a></i>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_2_1_1"/>
<font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:48_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5" target="_self">div</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:48_1_1_2"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:48_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E3:48_1_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:48_1_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:48_1_1_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="nat_1.html#E3:48_1_1_2"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E11:48_1_1_2"/><b>then </b>
<font color="Maroon">l</font> <a href="nat_1.html#K8" target="_self">lcm</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E12:48_1_1_2"/>
<font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">n</font> holds <br/><font color="Olive">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="nat_1.html#E2:48_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>


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