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

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> implies  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">and </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">y</font></span><br/>

<a NAME="E3:87"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="square_1.html#D4" target="_self" title="SQUARE_1:def.4">Def4</a></span>;<br/>
<a NAME="E4:87"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="square_1.html#D4" target="_self" title="SQUARE_1:def.4">Def4</a></span>;<br/>
<a NAME="E5:87"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="square_1.html#D4" target="_self" title="SQUARE_1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:87"/>
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="square_1.html#T77" target="_self" title="SQUARE_1:th.77">Th77</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
