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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:125"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="square_1.html#K5">^2</a> 
 ;<br/>

<a NAME="E2:125"/>
<font color="Maroon">b</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/>
<a NAME="E3:125"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1"><b>suppose </b></a><a NAME="E1:125_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:125_1_1"/>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_2"><b>suppose </b></a><a NAME="E1:125_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:125_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>;<br/><a NAME="E3:125_1_2"/>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><b>hence </b><a NAME="E4:125_1_2"/>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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