<?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="E1:12"/><i><font color="Green">E12</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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="bhsp_3.html#NR5" target="_self">is_Hilbert_space</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="bhsp_5.html#M2">OrthonormalFamily</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:12"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</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>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E4:12"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_6.html#V1">summable_set</a>
 ;<br/>

<a NAME="E5:12"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="bhsp_7.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">b<sub>1</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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E18</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:12_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:12_1"/>
<font color="Maroon">c<sub>4</sub></font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M2">OrthonormalFamily</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="bhsp_7.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M1">OrthogonalFamily</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_5.html#D9">BHSP_5:def 9</a></i>;<br/>
<a NAME="E5:12_1"/>
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>4</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:12_1"/>
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">c<sub>4</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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="bhsp_7.html#E2:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="bhsp_7.html#E4:12_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="bhsp_7.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E7:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E18</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:12_2"/><i><font color="Green">E19</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:12_2"/><i><font color="Green">E20</font></i>: 
( not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_7.html#E4:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="bhsp_7.html#T7" target="_self">Th7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:12_2"/><i><font color="Green">E21</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">b<sub>1</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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E5:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bhsp_7.html#E2:12_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bhsp_6.html#D7">BHSP_6:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="bhsp_5.html#K5">setopfunc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E8:12_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="bhsp_7.html#E6:12_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E9:12_2"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E6:12_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E10:12_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E11:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> 
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E6:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="bhsp_7.html#E8:12_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E12:12_2"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="bhsp_7.html#E8:12_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E13:12_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </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="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:12_2"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E15:12_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E9:12_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="bhsp_7.html#E12:12_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E16:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,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>3</sub></font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/>
<b>hence </b><a NAME="E17:12_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E14:12_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<span class="p1">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K2">sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_6.html#K3">sum_byfunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E11"><i><font color="Green">Lemma11</font></i></a></i>;<br/>


</div>
