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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">n1</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n2</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>set </b><font color="Maroon">s</font> =  <a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">s2</font> =  <a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">n1</font>, <font color="Olive">n2</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">n1</font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">n2</font> <a href="hidden.html#R1">=</a> 1;<br/>
<a NAME="E1:82"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n2</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 )
 ;<br/>

<a NAME="E2:82_1"/>
<font color="Maroon">n1</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/>
<a NAME="E3:82_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K10">hcf</a> 1 <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="E4:82_1"/>
<font color="Maroon">n1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:82"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n1</font>, <font color="Olive">n2</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> holds <br/><font color="Olive">n1</font> <a href="nat_1.html#K10">hcf</a> <font color="Olive">n2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> 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="real_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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="real_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:82_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n3</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> 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="real_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:82_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n4</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E8:82_2"/>
<font color="Maroon">n3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:82_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:82"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="real_3.html#D2" target="_self">Def2</a>, );<br/></i>
<b>hence </b><a NAME="E4:82"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#NK10" target="_self">c_d</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> implies <font color="Maroon">n1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>


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