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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:31"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="series_3.html#T6">SERIES_3:6</a></i>;<br/>
<a NAME="E2:31"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green">E37</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K3">^2</a> 
 
;<br/>
<a NAME="E4:31"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E44</font></i>: 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="series_5.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</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="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="series_5.html#E2"><i><font color="Green">Lemma32</font></i></a>, <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="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_2"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a>, <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_3"><b>suppose </b></a><a NAME="E1:31_1_3"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_3"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="series_5.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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