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<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:37"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:37"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:37"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <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="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:37"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:37"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="square_1.html#T96">SQUARE_1:96</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E10:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
;<br/>
<a NAME="E11:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E12:37"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
;<br/>
<a NAME="E13:37"/><i><font color="Green">E72</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:37"/>
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E15:37"/><b>then </b>
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:37"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E17:37"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E18:37"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E19:37"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E20:37"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E21:37"/>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E22:37"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E23:37"/>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span>
 <b>by </b><i>, , <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>


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