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<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="firebrick">::  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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>) )</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="firebrick">::  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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>) )</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="firebrick">::  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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="Olive" title="b1">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="Olive" title="b1">k</font>)</span>)</span> <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="Olive" title="b1">k</font>) )</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 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">thus </span><a NAME="E1:45"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</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="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">Th23</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 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/>

<a NAME="E2:45"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( <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> <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> &amp; <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> <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>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">Th23</a>, <a class="ref" href="lpspace1.html#T28" title="LPSPACE1:th.28">LPSPACE1:28</a></span>;<br/>
<a NAME="E3:45"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K7" title="LPSPACE1:func.7">RLSp_PFunct</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">Th23</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:45"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 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="#E2:45"><span class="lab"><font color="Green" title="E38">A1</font></span></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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