<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E13</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M2" target="_self">OrthonormalFamily</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:24"/><i><font color="Green">E14</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E1:24_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bhsp_5.html#E1:24_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:24_1_1"/>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bhsp_5.html#E1:24_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E6:24_1_1"/>
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:24_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E2:24_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="bhsp_5.html#E1:24_1"><i><font color="Green">E16</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E5:24_1"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</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="E1:24_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:24_1"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><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>6</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:24_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E5:24_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E9:24_1"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_5.html#E2:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E3:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E8:24_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_5.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E10:24_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_5.html#E2:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E3:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E8:24_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_5.html#T12" target="_self">Th12</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></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#K6">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T18">BHSP_1:18</a></i>
<br/>.= 
<span class="p1">(<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#K10">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_5.html#E2:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E3:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E8:24_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_5.html#T12" target="_self">Th12</a></i>
;<br/><b>hence </b><a NAME="E12:24_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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#K10">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:24"/><b>then </b><i><font color="Green">E16</font></i>: 
0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:24"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <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></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E5:24"><i><font color="Green">E16</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:24"/>
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">c<sub>4</sub></font>,<a href="binop_2.html#K33">addreal</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><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> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/>


</div>
