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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="bhsp_3.html#NR5" title="BHSP_3:NR.5">is_Hilbert_space</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c2">S</font> be    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green" title="E13">A2</font></i>: 
not <font color="Maroon" title="c2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E3:12"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">H</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font> ) )
 ;<br/>

<b>assume </b><a NAME="E4:12"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="bhsp_6.html#V1" title="BHSP_6:attr.1">summable_set</a>
 ;<br/>

<a NAME="E5:12"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="bhsp_7.html#T6" target="_self" title="BHSP_7:th.6">Th6</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E17">A6</font></i>: 
for <font color="Olive" title="b1">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b1">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/><span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b1">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b1">Y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b1">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">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" title="c4">Y1</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E18">A7</font></i>: 
( not <font color="Maroon" title="c4">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 ;<br/>

<a NAME="E2:12_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:12_1"/>
<font color="Maroon" title="c4">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="bhsp_7.html#T5" target="_self" title="BHSP_7:th.5">Th5</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c4">Y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>;<br/>
<a NAME="E5:12_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y1</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green" title="E18">A7</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" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c4">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c4">Y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c4">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E1:12_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="bhsp_7.html#E2:12_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="bhsp_7.html#E4:12_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="bhsp_7.html#T3" target="_self" title="BHSP_7:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c4">p1</font> = <span class="p1">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">p2</font> =  <a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E7:12"/>
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</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" title="c6">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E18">A10</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font>
 ;<br/>

<a NAME="E2:12_2"/><i><font color="Green" title="E19">A11</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:12_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">Y01</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:12_2"/><i><font color="Green" title="E20">A12</font></i>: 
( not <font color="Maroon" title="c7">Y01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">Y01</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c7">Y01</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y1</font> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b1">Y1</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b1">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E4:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="bhsp_7.html#T7" target="_self" title="BHSP_7:th.7">Th7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">Y02</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:12_2"/><i><font color="Green" title="E21">A13</font></i>: 
( not <font color="Maroon" title="c8">Y02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">Y02</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">Y1</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c8">Y02</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y1</font> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Olive" title="b1">Y1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) )
 <b>by </b><i><a class="txt" href="bhsp_7.html#E5:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="bhsp_7.html#E2:12_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="bhsp_6.html#D7" title="BHSP_6:def.7">BHSP_6:def 7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">Y1</font> = <font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font>;<br/>
<b>set </b><font color="Maroon" title="c10">r</font> =  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Y011</font> = <font color="Maroon" title="c7">Y01</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/>
<a NAME="E8:12_2"/><i><font color="Green" title="E22">A14</font></i>: 
( <font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">Y01</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font> &amp; <font color="Maroon" title="c8">Y02</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font> &amp; <font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="bhsp_7.html#E6:12_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E9:12_2"/><b>then </b><i><font color="Green" title="E23">A15</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E6:12_2"><i><font color="Green" title="E21">A13</font></i></a></i>;<br/>
<a NAME="E10:12_2"/>
<font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Y011</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">Y02</font>
 
;<br/>
<a NAME="E11:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E6:12"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="bhsp_7.html#E8:12_2"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>
<a NAME="E12:12_2"/><b>then </b><i><font color="Green" title="E24">A16</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E4:12_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="bhsp_7.html#E8:12_2"><i><font color="Green" title="E22">A14</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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:12_2"/><b>then </b><i><font color="Green" title="E25">A17</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142" title="COMPLEX1:th.142">COMPLEX1:142</a></i>;<br/>
<a NAME="E15:12_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E9:12_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="bhsp_7.html#E12:12_2"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">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" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">Y01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Y02</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/>
<b>hence </b><a NAME="E17:12_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E14:12_2"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K2" title="BHSP_6:func.2">sum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_6.html#K3" title="BHSP_6:func.3">sum_byfunc</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E11"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/>


</div>
