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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  implies  for <font color="Olive" title="b1">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y</font> is  <a href="bhsp_6.html#V3" title="BHSP_6:attr.3">weakly_summable_set</a>  holds <br/> for <font color="Olive" title="b2">L</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  holds <br/><font color="Olive" title="b1">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Olive" title="b2">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
(  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y</font> is  <a href="bhsp_6.html#V3" title="BHSP_6:attr.3">weakly_summable_set</a>  holds <br/> for <font color="Olive" title="b2">L</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  holds <br/><font color="Olive" title="b1">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Olive" title="b2">L</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Y</font> is  <a href="bhsp_6.html#V3" title="BHSP_6:attr.3">weakly_summable_set</a>  implies  for <font color="Olive" title="b1">L</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  holds <br/><font color="Maroon" title="c2">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Olive" title="b1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:14"/>
<font color="Maroon" title="c2">Y</font> is  <a href="bhsp_6.html#V3" title="BHSP_6:attr.3">weakly_summable_set</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  holds <br/><font color="Maroon" title="c2">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Olive" title="b1">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 for <font color="Olive" title="b1">L</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  holds <br/> for <font color="Olive" title="b2">e</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b3">Y0</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 <br/>(  not <font color="Olive" title="b3">Y0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b3">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b4">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="Olive" title="b3">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Y1</font> &amp; <font color="Olive" title="b4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</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"><font color="Olive" title="b1">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> (<font color="Olive" title="b4">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="Olive" title="b1">L</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="Olive" title="b2">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="bhsp_6.html#T6" target="_self" title="BHSP_6:th.6">Th6</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">L</font> be   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>  implies <font color="Maroon" title="c2">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c4">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:14"/>
<font color="Maroon" title="c4">L</font> is  <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c4">L</font></span><br/>

<a NAME="E6:14"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">Y0</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 <br/>(  not <font color="Olive" title="b2">Y0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b3">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="Olive" title="b2">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Y1</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</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"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> (<font color="Olive" title="b3">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="c4">L</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="Olive" title="b1">e</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:14"/>
<font color="Maroon" title="c2">Y</font> <a href="bhsp_6.html#R1" title="BHSP_6:pred.1">is_summable_set_by</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_6.html#D6" target="_self" title="BHSP_6:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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