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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">n</font>;<br/>
<a NAME="E1:2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:2_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E3:2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:2_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E3:2_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3"><b>suppose </b></a><a NAME="E1:2_1_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:2_1_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/><b>thus </b><a NAME="E3:2_1_1_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_3_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_3_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E3:2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_3_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_3_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">c</font> = <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>;<br/><a NAME="E2:2_1_1_3_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E3:2_1_1_3_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E4:2_1_1_3_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E5:2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><a NAME="E6:2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:2_1_1_3_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="wsierp_1.html#T20">WSIERP_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:2_1_1_3_1_1_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:2_1_1_3_1_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E5:2_1_1_3_1_1_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><a NAME="E6:2_1_1_3_1_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_3_1_1_3_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_3_1_1_3_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:2_1_1_3_1_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">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="E10:2_1_1_3_1_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_3_1_1_3_2"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E2:2_1_1_3_1_1_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:2_1_1_3_1_1_3_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/>
<a NAME="E4:2_1_1_3_1_1_3_2"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:2_1_1_3_1_1_3_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E6:2_1_1_3_1_1_3_2"/><b>then </b>
<font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E7:2_1_1_3_1_1_3_2"/><b>then </b>
<font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:2_1_1_3_1_1_3_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_1_1_3_1_1_3_2"/>
<font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E10:2_1_1_3_1_1_3_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:2_1_1_3_1_1_3"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1_1_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_3_1_1_3_3"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:2_1_1_3_1_1_3_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E3:2_1_1_3_1_1_3_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T20">WSIERP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_1_3_1_1_3_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:2_1_1_3_1_1_3"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1_1_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:2_1_1_3_1_1_3_4"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_3_1_1_3_4"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:2_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:2_1_1_3_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">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="E17:2_1_1_3_1_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1_1_3_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c</font> = <font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font>;<br/>
<a NAME="E1:2_1_1_3_1_1_3_5"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E2:2_1_1_3_1_1_3_5"/>
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E3:2_1_1_3_1_1_3_5"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E4:2_1_1_3_1_1_3_5"/>
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E5:2_1_1_3_1_1_3_5"/><b>then </b>
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:2_1_1_3_1_1_3_5"/>
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20">WSIERP_1:20</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E52</font></i>: <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pythtrip.html#T8">PYTHTRIP:8</a></i>
;<br/><a NAME="E19:2_1_1_3_1_1_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pythtrip.html#T8">PYTHTRIP:8</a></i>;<br/><b>hence </b><a NAME="E20:2_1_1_3_1_1_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<b>hence </b><a NAME="E3:2"/>
for <font color="Olive">k</font>, <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">k</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">n</font>
 ;<br/>


</div>
