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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:33"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</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">.||</a></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">.|.</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="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/>
<a NAME="E2:33"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</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">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E26</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">.|.</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><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</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">.||</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="E5:33"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E7:33"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</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">.||</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">.|.</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/>
<b>then </b><span class="p1"><a href="bhsp_1.html#K3">||.</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">.||</a></span> <a href="square_1.html#K5">^2</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">.|.</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="txt" href="rusub_5.html#E1:33"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="square_1.html#T96">SQUARE_1:96</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2">.|.</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T21">BHSP_1:21</a></i>
<br/>.= 
<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">.|.</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_5.html#E2:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</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">.|.</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">.|.</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="txt" href="rusub_5.html#E3:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E9:33"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</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">.||</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">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</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">.|.</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">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/>

<a NAME="E10:33"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></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#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/>
<a NAME="E11:33"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/>
<a NAME="E12:33"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E13:33"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</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="E14:33"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</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#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rusub_5.html#T2" target="_self">Th2</a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<b>then </b><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T23">BHSP_1:23</a></i>
<br/>.= 
<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">.|.</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#K5">-</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_5.html#E2:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</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">.|.</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">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</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">.|.</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">.|.</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="txt" href="rusub_5.html#E3:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E16:33"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</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">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</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">.|.</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">||.</a><span class="default"><font color="Maroon">y</font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/>


</div>
