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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  implies  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:123"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 ; <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#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E2:123"/>
<font color="Maroon" title="c2">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E3:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A2</font></span>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:123_1"/>
( <font color="Maroon" title="c2">b</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>  or <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1"><span class="kw">suppose </span></a><a NAME="E1:123_1_1"/>
<font color="Maroon" title="c2">b</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> 
 ; <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#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:123_1_1"/>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:123"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_2"><span class="kw">suppose </span></a><a NAME="E1:123_1_2"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font></span><br/><div class="add"><a NAME="E2:123_1_2"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:123"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="square_1.html#T90" title="SQUARE_1:th.90">SQUARE_1:90</a></span>;<br/><span class="kw">hence </span><a NAME="E3:123_1_2"/>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123_1_2"><span class="lab"><font color="Green" title="E70">A3</font></span></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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