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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</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="E1:10"/>
<font color="Maroon">Y</font> is <a href="bhsp_6.html#V1" target="_self">summable_set</a>
 ;<br/>

<b>then 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="E3:10"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> 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/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Olive">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><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="E1:10_1"/><i><font color="Green">E33</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">K</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E34</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">K</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds   <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Olive">x</font></span><a href="bhsp_1.html#K3">.||</a></span> ) )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font>
 ;<br/><b>set </b><font color="Maroon">e</font> = <font color="Maroon">e1</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">K</font>;<br/><a NAME="E2:10_1_1"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E3:10_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">K</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</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="E5:10_1_1"/><i><font color="Green">E43</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">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="Maroon">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/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Olive">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">K</font> ) )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_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="E1:10_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E2:10_1_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:10_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E4:10_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">e1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">e1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Maroon">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:10_1_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">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"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Olive">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e1</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:10_1"/>
for <font color="Olive">e1</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e1</font> 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"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" target="_self">setsum</a> <font color="Olive">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e1</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon">Y</font> is <a href="bhsp_6.html#V3" target="_self">weakly_summable_set</a>
 <b>by </b><i/>;<br/>


</div>
