<?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="E1:23"/><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="E2:23"/><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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:23"/><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="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</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="E4:23"/><i><font color="Green">E16</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>5</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>5</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/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of 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="E6:23"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <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>4</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:23"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_5.html#E3:23"><i><font color="Green">E15</font></i></a>, <a class="ref" href="bhsp_5.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E1:23_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E2:23_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="bhsp_5.html#M1" target="_self">OrthogonalFamily</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:23_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E1:23_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_5.html#D8" target="_self">Def8</a></i>;<br/>
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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
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"><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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E3:23"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E1:23_1"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
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"><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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<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>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E3:23"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E1:23_1"><i><font color="Green">E19</font></i></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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <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>8</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</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>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="bhsp_1.html#K2">.|.</a> <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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8">BHSP_1:8</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>8</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8">BHSP_1:8</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="bhsp_5.html#E3:23_1"><i><font color="Green">E20</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:23_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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:23"/><i><font color="Green">E19</font></i>: 
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>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="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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:23_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:23_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E3:23"><i><font color="Green">E15</font></i></a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<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>7</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bhsp_5.html#E1:23_2"><i><font color="Green">E20</font></i></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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</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>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 1

<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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</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>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E1:23_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="bhsp_5.html#D9" target="_self">Def9</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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>

<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>7</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8">BHSP_1:8</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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <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>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T8">BHSP_1:8</a></i>
<br/>.= 
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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E2:23_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:23_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<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>4</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>4</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
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"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E6:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E2:23"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E3:23"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E4:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bhsp_5.html#E6:23"><i><font color="Green">E17</font></i></a>, <a class="txt" href="bhsp_5.html#E7:23"><i><font color="Green">E18</font></i></a>, <a class="txt" href="bhsp_5.html#E8:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_5.html#T9" target="_self">Th9</a></i>
<br/>.= 
 <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>5</sub></font>,<a href="binop_2.html#K33">addreal</a> 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bhsp_5.html#E6:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="bhsp_5.html#D5" target="_self">Def5</a></i>
;<br/>
<b>hence </b><a NAME="E10:23"/>
<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>4</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>4</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>  <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>5</sub></font>,<a href="binop_2.html#K33">addreal</a> 
 ;<br/>


</div>
