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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font>;<br/>
<b>consider </b><font color="Maroon">N</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green">E24</font></i>: 
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">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>consider </b><font color="Maroon">M</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green">E25</font></i>: 
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">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<b>consider </b><font color="Maroon">K</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:9_1"/><i><font color="Green">E28</font></i>: 
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">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 3;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:9_1"/><i><font color="Green">E29</font></i>: 
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">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 4;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:9_1"/><i><font color="Green">E34</font></i>: 
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">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> iff <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>) ) );<br/>
<a NAME="E11:9_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">n</font>,<font color="Olive">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<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>consider </b><font color="Maroon">m</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="E2:9_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>);<br/><a NAME="E2:9_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i/>;<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:9_1_1_1_1_1_1"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:9_1_1_1_1_1_2"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:9_1_1_1_1_1_3"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><a NAME="E2:9_1_1_1_1_1_4"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:9_1_1_1_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>);<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:9_1_1_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E4:9_1_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:9_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:9_1_1_1_1_2_2"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_2_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:9_1_1_1_1_2_3"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_2_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><a NAME="E2:9_1_1_1_1_2_4"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_1_1_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:9_1_1_1_1_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>);<br/><div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:9_1_1_1_1_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E4:9_1_1_1_1_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:9_1_1_1_1_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_3_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:9_1_1_1_1_3_2"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_3_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_3_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:9_1_1_1_1_3_3"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_3_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_3_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_3_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><a NAME="E2:9_1_1_1_1_3_4"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_3_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_1_1_1_1_3"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:9_1_1_1_1_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>);<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_4_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:9_1_1_1_1_4_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E4:9_1_1_1_1_4_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:9_1_1_1_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_4_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:9_1_1_1_1_4_2"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_4_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:9_1_1_1_1_4_3"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_4_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_4_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><a NAME="E2:9_1_1_1_1_4_4"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_4_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_1_1_1_1_4"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:9_1_1_1_1_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>);<br/><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_5_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">k</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="E3:9_1_1_1_1_5_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E4:9_1_1_1_1_5_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:9_1_1_1_1_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_5_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:9_1_1_1_1_5_2"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_5_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_5_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:9_1_1_1_1_5_3"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_5_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_5_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_1_5_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:9_1_1_1_1_5_4"/><b>then </b>
 ex <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="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1_1_5_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_1_1_1_1_5"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">n</font>,<font color="Olive">r</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:9_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<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/>

<a NAME="E15:9_1"/>
5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)

;<br/><br/>

<a NAME="E17:9_1"/>
<span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>)

;<br/><br/>

<a NAME="E19:9_1"/>
<span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">n</font>)

;<br/><br/>

<a NAME="E21:9_1"/>
<span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">n</font>)

;<br/><br/>

<a NAME="E23:9_1"/>
<span class="p1">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">5 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K7">/</a> 5</span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">n</font>)

;<br/><br/>


</div>
