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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:10_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E4:10_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i><a class="txt" href="nat_3.html#E1:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E5:10_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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="E6:10_1"/>
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="E5:10"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E6:10"/><b>then </b><i><font color="Green">E9</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:10"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E7:10"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:10_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1
;<br/>

<a NAME="E3:10_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:10_2"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:10_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:10_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E6:10_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E7:10_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E8:10_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E9:10_2"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E4:10_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E10:10_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E2:10_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E11:10_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1]
 <b>by </b><i><a class="txt" href="nat_3.html#E2:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="nat_3.html#E9:10_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:10"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7">NAT_1:sch 7</a>(<a class="txt" href="nat_3.html#E6:10"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nat_3.html#E7:10"><i><font color="Green">E10</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:10"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E2:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>


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