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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> iff <font color="Maroon">a<sub>1</sub></font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> <a href="pepin.html#K2">^2</a>  );<br/>
<a NAME="E1:19"/><i><font color="Green">E30</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 ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> iff <font color="Olive">k</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> <a href="pepin.html#K2">^2</a>  )
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">k'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:19_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</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="E4:19_1_1"/>
<font color="Maroon">n</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k'</font> <a href="pepin.html#K2">^2</a> </span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:19_1_1"/>
<font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:19_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</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="19_1_2_1"><b>suppose </b></a><a NAME="E1:19_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:19_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="pepin.html#K2">^2</a>  <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/><a NAME="E4:19_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E5:19_1_2_1"/>
<font color="Maroon">m</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#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_2"><b>suppose </b></a><a NAME="E1:19_1_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_2_2"/>
<font color="Maroon">m</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#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_3"><b>suppose </b></a><a NAME="E1:19_1_2_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:19_1_2_3"/><b>then </b>
<font color="Maroon">m</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/><b>then consider </b><font color="Maroon">p'</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="E4:19_1_2_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p'</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p'</font> as  <a href="int_2.html#V1">prime</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">m'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:19_1_2_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m'</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">m'</font> = <font color="Maroon">m'</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="E9:19_1_2_3"/>
<font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E10:19_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:19_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><a NAME="E12:19_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E13:19_1_2_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">n'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:19_1_2_3"/><i><font color="Green">E43</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">n'</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">n'</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="E17:19_1_2_3"/>
<font color="Maroon">m'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E18:19_1_2_3"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E19:19_1_2_3"/><i><font color="Green">E45</font></i>: 
<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="E20:19_1_2_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">m'</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/><b>consider </b><font color="Maroon">k'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E22:19_1_2_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</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="E23:19_1_2_3"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k'</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E24:19_1_2_3"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E25:19_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="pepin.html#K2">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E26:19_1_2_3"/><b>then </b>
<font color="Maroon">n'</font> <a href="pepin.html#K2">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="pepin.html#K2">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E27:19_1_2_3"/><b>then </b>
<font color="Maroon">m'</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">n'</font> <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E28:19_1_2_3"/><b>then </b>
<font color="Maroon">m'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n'</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E30:19_1_2_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> <a href="xcmplx_0.html#K3">*</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="E31:19_1_2_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E32:19_1_2_3"/>
<font color="Maroon">m</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#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:19"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</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="E3:19"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> iff <font color="Maroon">m</font> <a href="pepin.html#K2">^2</a>  <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="pepin.html#K2">^2</a>  )
 ;<br/>


</div>
