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<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>take </b><font color="Maroon">w</font> = 0;<br/><b>thus </b><a NAME="E2:46_1_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">w</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">w</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E3:46_1_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> )
 ;<br/><b>hence </b><a NAME="E4:46_1_1_1"/>
<font color="Maroon">w</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font>, <font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:46_1_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">k1</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k1</font> <a href="nat_1.html#K2" target="_self">*</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">k1</font> <a href="nat_1.html#K2" target="_self">*</a> <font color="Maroon">n1</font> )
 <b>by </b><i/>;<br/><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">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E4:46_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E5:46_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:46_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E7:46_1_1_2"/><b>then </b><i><font color="Green">E38</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>]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:46_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>consider </b><font color="Maroon">l</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="E9:46_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">l</font>] &amp; ( 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="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>();<br/></i><b>take </b><font color="Maroon">l</font> = <font color="Maroon">l</font>;<br/><b>thus </b><a NAME="E10:46_1_1_2"/>
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E11:46_1_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:46_1_1_2_1"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E2:46_1_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="nat_1.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:46_1_1_2_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:46_1_1_2_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T4">XCMPLX_1:4</a></i>;<br/><a NAME="E5:46_1_1_2_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">l</font> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">l</font> <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E6:46_1_1_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="nat_1.html#E1:46_1_1_2"><i><font color="Green">E34</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:46_1_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E3:46_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="txt" href="nat_1.html#E7:46_1_1_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E4:46_1_1_2_1_1"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">l</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:46_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_1.html#E3:46_1_1_2"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:46_1_1_2_1"/>
<font color="Maroon">l</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:46_1_1_2"/>
<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>
<b>hence </b><a NAME="E3:46_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">k</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">m</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">m</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Olive">m</font> ) )
 <b>by </b><i><a class="ref" href="nat_1.html#T2" target="_self">Th2</a></i>;<br/>


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