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<div class="add">

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

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E38</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">X</font> <a href="bhsp_3.html#NR5" target="_self">is_Hilbert_space</a>  )
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="bhsp_5.html#M2">OrthonormalFamily</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E2:9"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">x</font> ) )
 ;<br/>

<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S</font> is <a href="bhsp_6.html#V1">summable_set</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E56</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>set </b><font color="Maroon">e'</font> =  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">e</font>;<br/><a NAME="E2:9_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><b>then consider </b><font color="Maroon">e1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E57</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T1">CHAIN_1:1</a></i>;<br/><a NAME="E5:9_1_1"/>
<font color="Maroon">e1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">e</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E6:9_1_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">e</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E7:9_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">e1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">e1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i>, , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:9_1_1"/><i><font color="Green">E60</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font> ) )
 <b>by </b><i>, , , <a class="ref" href="bhsp_6.html#T10">BHSP_6:10</a></i>;<br/><b>take </b><font color="Maroon">Y0</font> = <font color="Maroon">Y0</font>;<br/><b>thus </b><a NAME="E10:9_1_1"/>
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E11:9_1_1"/><i><font color="Green">E62</font></i>: 
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y1</font> )
 ;<br/><a NAME="E12:9_1_1"/><i><font color="Green">E64</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/><a NAME="E13:9_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:9_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E15:9_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e1</font>
 ;<br/><a NAME="E16:9_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:9_1_1"/>
<font color="Maroon">Y1</font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M2">OrthonormalFamily</a> of <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:9_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">Y1</font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M1">OrthogonalFamily</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="bhsp_5.html#D9">BHSP_5:def 9</a></i>;<br/><a NAME="E19:9_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E20:9_1_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E21:9_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</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">Y1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , , , </i>;<br/><a NAME="E22:9_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/><a NAME="E23:9_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</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">Y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/><a NAME="E24:9_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E25:9_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, <a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/><b>hence </b><a NAME="E26:9_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon">S</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">H</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:9_2_1"/><i><font color="Green">E78</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>set </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font>;<br/><a NAME="E2:9_2_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><b>then consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:9_2_1"/><i><font color="Green">E79</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font> ) )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:9_2_1_1"/><i><font color="Green">E80</font></i>: 
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y1</font> )
 ;<br/><b>set </b><font color="Maroon">F</font> =  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> ;<br/><a NAME="E2:9_2_1_1"/><i><font color="Green">E81</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:9_2_1_1"/>
 <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E4:9_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:9_2_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E6:9_2_1_1"/>
<font color="Maroon">Y1</font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M2">OrthonormalFamily</a> of <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:9_2_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">Y1</font> is  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M1">OrthogonalFamily</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="bhsp_5.html#D9">BHSP_5:def 9</a></i>;<br/><a NAME="E8:9_2_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:9_2_1_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:9_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</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">Y1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , , , </i>;<br/><a NAME="E11:9_2_1_1"/><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</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">Y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/><a NAME="E12:9_2_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D4">BHSP_1:def 4</a></i>;<br/><a NAME="E13:9_2_1_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E14:9_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E15:9_2_1_1"/><i><font color="Green">E88</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/><a NAME="E16:9_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E17:9_2_1_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E18:9_2_1_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><b>hence </b><a NAME="E19:9_2_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_2_1"/>
 ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Olive">Y1</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_2"/>
<font color="Maroon">S</font> is <a href="bhsp_6.html#V1">summable_set</a>
 <b>by </b><i>, <a class="ref" href="bhsp_6.html#T10">BHSP_6:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
( <font color="Maroon">S</font> is <a href="bhsp_6.html#V1">summable_set</a> iff <font color="Maroon">S</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>


</div>
