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

<b>let </b><font color="Maroon" title="c6">a3</font>, <font color="Maroon" title="c7">b3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green" title="E36">A3</font></i>: 
(  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> ) &amp;  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">a01</font>, <font color="Maroon" title="c9">a02</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:67_1_2"/><i><font color="Green" title="E37">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">a01</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">a02</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a01</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a02</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">a2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:67_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c11">a2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b3</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:67_1_2"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a001</font>, <font color="Maroon" title="c13">a002</font>, <font color="Maroon" title="c14">a003</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:67_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a001</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">a002</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a003</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">a001</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c13">a002</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c14">a003</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b1">b01</font>, <font color="Olive" title="b2">b02</font>, <font color="Olive" title="b3">b03</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">b01</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b02</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b03</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">b01</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b2">b02</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b3">b03</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>( <font color="Olive" title="b1">b01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a001</font> &amp; <font color="Olive" title="b2">b02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a002</font> &amp; <font color="Olive" title="b3">b03</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a003</font> ) ) )
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:67"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="euclid_3.html#T60" target="_self" title="EUCLID_3:th.60">Th60</a></i>;<br/>
<a NAME="E8:67_1_2"/>
( <font color="Maroon" title="c8">a01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a001</font> &amp; <font color="Maroon" title="c9">a02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a002</font> &amp; <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a003</font> )
 
<b>by </b><i><a class="txt" href="euclid_3.html#E3:67_1_2"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="euclid_3.html#E7:67_1_2"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:67_1_2"/>
<font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b3</font>
 <b>by </b><i><a class="txt" href="euclid_3.html#E5:67_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="euclid_3.html#E7:67_1_2"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/>


</div>
