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<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="E1:13"/><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>  )
 ;<br/>

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E2:13"/>
<font color="Maroon" title="c2">Y</font> is <a href="bhsp_6.html#V3" title="BHSP_6:attr.3">weakly_summable_set</a>
 ;<br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green" title="E13">A2</font></i>: 
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> 0 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"><font color="Olive" title="b1">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Olive" title="b4">Y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">e</font> ) )
 <b>by </b><i><a class="ref" href="bhsp_6.html#D5" target="_self" title="BHSP_6:def.5">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c4">L</font> is <a href="bhsp_6.html#V2" title="BHSP_6:attr.2">Bounded</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
( not <font color="Maroon" title="c6">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c6">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="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="c6">Y0</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">Y</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_6.html#E4:13"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="bhsp_6.html#E1:13_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="bhsp_6.html#E1:13_1_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c7">Y0</font> = <font color="Maroon" title="c6">Y0</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="E1:13_1_1_1"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c7">Y0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y1</font> &amp; <font color="Maroon" title="c8">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/><b>set </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c8">Y1</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c3">x</font>;<br/><b>set </b><font color="Maroon" title="c11">y1</font> =  <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c8">Y1</font>;<br/><a NAME="E2:13_1_1_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c8">Y1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c8">Y1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T32" title="HAHNBAN:th.32">HAHNBAN:32</a></i>;<br/><a NAME="E3:13_1_1_1"/>
<font color="Maroon" title="c8">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="bhsp_6.html#E3:13_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="bhsp_6.html#E1:13_1_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c8">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="c8">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> 
 <b>by </b><i><a class="txt" href="bhsp_6.html#E1:13"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="bhsp_6.html#T5" target="_self" title="BHSP_6:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E5:13_1_1_1"/>
 <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="Maroon" title="c8">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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="bhsp_6.html#E3:13_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="bhsp_6.html#E1:13_1_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="bhsp_6.html#E2:13_1_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:13_1_1"/>
( not <font color="Maroon" title="c7">Y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">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="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">Y0</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">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="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="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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_6.html#E3:13_1_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:13_1"/>
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> 0 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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
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> 0 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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">e</font> ) )
 ;<br/>


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