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<b>let </b><font color="Maroon">p</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">i</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="E1:46"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<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>(  not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Olive">k</font> );<br/>
<a NAME="E3:46"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 0
 ;<br/><a NAME="E2:46_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:46_1_1"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>thus </b><a NAME="E5:46_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:46"/><i><font color="Green">E51</font></i>: 
for <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">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<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="E1:46_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">u1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:46_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:46_2_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u1</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E5:46_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1_1_1"/>
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">i</font> or <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u1</font> )
 </span><b>by </b><i>, , <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:46_2_1_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:46_2_1_1_1_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:46_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_1_1_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">u1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:46_2_1_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E4:46_2_1_1_1_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E5:46_2_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/><a NAME="E6:46_2_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">w2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/><b>then consider </b><font color="Maroon">w3</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="E8:46_2_1_1_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">w3</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>thus </b><a NAME="E9:46_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:46_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:46_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:46"/>
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:46"/>
(  not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Olive">k</font> )
 ;<br/>


</div>
