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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:17"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="bhsp_1.html#R1">are_orthogonal</a> 
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D3">BHSP_1:def 3</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></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="E4:17"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></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="E5:17"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></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/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></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"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<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"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E3:17"><i><font color="Green">E8</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"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></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"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E2:17"><i><font color="Green">E7</font></i></a>, <a class="txt" href="bhsp_5.html#E4:17"><i><font color="Green">E9</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"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</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">c<sub>3</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E5:17"><i><font color="Green">E10</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"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</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">c<sub>3</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</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/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></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/>
<b>hence </b><a NAME="E7:17"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></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"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 ;<br/>


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