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

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 2 <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">a</font> )
 ;<br/>

<i><font color="Green" title="E32">A2</font></i>: <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 = 
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
<b>by </b><i><a class="ref" href="newton.html#T100" title="NEWTON:th.100">NEWTON:100</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>
;<br/>
<a NAME="E3:38"/>
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:38"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="prepower.html#E2:38"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="prepower.html#T28" target="_self" title="PREPOWER:th.28">Th28</a></i>;<br/>


</div>
