<?xml version="1.0"?>
<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="E1:14"/><i><font color="Green">E29</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>  )
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Y</font> is <a href="bhsp_6.html#V3" target="_self">weakly_summable_set</a>
 ;<br/>

<b>let </b><font color="Maroon">L</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E3:14"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L</font> is <a href="bhsp_6.html#V2" target="_self">Bounded</a>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:14"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">L</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">X</font>  st <font color="Olive">L</font> is <a href="bhsp_6.html#V2" target="_self">Bounded</a> holds <br/>for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> 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">Y</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 <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<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="Olive">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E6:14"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> 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">Y</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 <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<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">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon">Y</font> <a href="bhsp_6.html#R1" target="_self">is_summable_set_by</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div>
