<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">Z</font> =  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">V</font> =  <a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> implies <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">v1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
( <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> &amp; <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c9">v1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">M</font> )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">u1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:43_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
( <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> &amp; <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c10">u1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>;<br/><a NAME="E6:43_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>;<br/><a NAME="E7:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T33" title="LPSPACE1:th.33">LPSPACE1:33</a></span>;<br/><a NAME="E8:43_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">v1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c9">v1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">u1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="lpspace1.html#T31" title="LPSPACE1:th.31">LPSPACE1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:43"/>
 <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> holds <br/><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> implies <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">u1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:43_2"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
( <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> &amp; <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c9">u1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">M</font> )
 ;<br/><a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E37">A10</font></span></a></span>;<br/><a NAME="E5:43_2"/>
<font color="Maroon" title="c7">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T33" title="LPSPACE1:th.33">LPSPACE1:33</a></span>;<br/><a NAME="E6:43_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c7">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">u1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="lpspace1.html#T32" title="LPSPACE1:th.32">LPSPACE1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_2"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:43"/>
 <a href="lpspace2.html#K3" title="LPSPACE2:func.3">AlmostZeroLpFunctions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="lpspace1.html#V1" title="LPSPACE1:attr.1">multi-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D1" title="LPSPACE1:def.1">LPSPACE1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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