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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:70"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:70_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:70"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:70_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E1:70_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>hence </b><a NAME="E4:70_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:70_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:70"><i><font color="Green">E51</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><a NAME="E3:70_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E1:70_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><b>hence </b><a NAME="E4:70_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_3"><b>suppose </b></a><a NAME="E1:70_1_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:70_1_1_3"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="ref" href="wsierp_1.html#T38" target="_self">Th38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="rat_1.html#NM1">Rational</a> ;<br/><font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="prepower.html#K7">#Z</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:70_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="wsierp_1.html#E3:70_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="prepower.html#T53">PREPOWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#T15">PREPOWER:15</a></i>
;<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="E7:70_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T32" target="_self">Th32</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E7:70_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
;<br/><a NAME="E9:70_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:70"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E10:70_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7:70_1_1_3"><i><font color="Green">E54</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="E4:70"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div>
