<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></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="E2:8_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></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="E4:8_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></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="E6:8_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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="E8:8_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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> 4</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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> 4</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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> 4</span>)</span>) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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> 4</span>)</span>) ) );<br/>
<a NAME="E9:8_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><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>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:8_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3 )
 <b>by </b><i><a class="ref" href="scheme1.html#T3" target="_self">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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="E1:8_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3 )
 </span><b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>);<br/><a NAME="E2:8_1_1_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_1"><i><font color="Green">E10</font></i></a></i>;<br/><div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scheme1.html#E4:8_1"><i><font color="Green">E5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="scheme1.html#E6:8_1"><i><font color="Green">E6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <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="8_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="txt" href="scheme1.html#E8:8_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:8_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="scheme1.html#E3:8_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E4:8_1_1_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>);<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_2_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E4:8_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="scheme1.html#E3:8_1_1_1_1_2_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:8_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="scheme1.html#E6:8_1"><i><font color="Green">E6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_2"><i><font color="Green">E10</font></i></a>, <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="8_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_2_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="txt" href="scheme1.html#E8:8_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:8_1_1_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_3"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>);<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_3_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E4:8_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="scheme1.html#E3:8_1_1_1_1_3_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:8_1_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_3_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_3_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scheme1.html#E4:8_1"><i><font color="Green">E5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_3"><i><font color="Green">E10</font></i></a>, <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="8_1_1_1_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_3_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_3_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i><a class="txt" href="scheme1.html#E8:8_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:8_1_1_1_1_3"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_4"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 3</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>);<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_4_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E4:8_1_1_1_1_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="scheme1.html#E3:8_1_1_1_1_4_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E5:8_1_1_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_4_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_4_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scheme1.html#E4:8_1"><i><font color="Green">E5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_4"><i><font color="Green">E10</font></i></a>, <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="8_1_1_1_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_4_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:8_1_1_1_1_4_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="scheme1.html#E6:8_1"><i><font color="Green">E6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scheme1.html#E1:8_1_1_1_1_4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T43">NAT_1:43</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="scheme1.html#E2:8_1_1_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:8_1_1_1_1_4"><i><font color="Green">E12</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="E4:8_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></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="E11:8_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="scheme1.html#E9:8_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E13:8_1"/>
4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scheme1.html#E2:8_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>)
<b>by </b><i><a class="txt" href="scheme1.html#E11:8_1"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>)

;<br/><br/>

<a NAME="E15:8_1"/>
<span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="scheme1.html#E4:8_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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> 4</span>)</span>)
<b>by </b><i><a class="txt" href="scheme1.html#E11:8_1"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>7</sub></font>)

;<br/><br/>

<a NAME="E17:8_1"/>
<span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="scheme1.html#E6:8_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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> 4</span>)</span>)
<b>by </b><i><a class="txt" href="scheme1.html#E11:8_1"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>7</sub></font>)

;<br/><br/>

<a NAME="E19:8_1"/>
<span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="scheme1.html#E8:8_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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">4 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></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> 4</span>)</span>)
<b>by </b><i><a class="txt" href="scheme1.html#E11:8_1"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)

;<br/><br/>


</div>
