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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="lpspace2.html#V1" title="LPSPACE2:attr.1">geq_than_1</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</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="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="lpspace2.html#V1" title="LPSPACE2:attr.1">geq_than_1</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="lpspace2.html#V1" title="LPSPACE2:attr.1">geq_than_1</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="lpspace2.html#V1" title="LPSPACE2:attr.1">geq_than_1</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="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">x</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span>;<br/>
<a NAME="E1:87"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</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="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span>
 
;<br/>
<a NAME="E2:87"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</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="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</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#D11" target="_self" title="LPSPACE2:def.11">Def11</a></span>;<br/>
<a NAME="E3:87"/><span class="kw">then </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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">Th38</a>, <a class="ref" href="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">Th23</a></span>;<br/>
<a NAME="E4:87"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<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></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T55" target="_self" title="LPSPACE2:th.55">Th56</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:87"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
( <font color="Maroon" title="c6">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<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></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1">(<span class="default"><a href="lpspace2.html#K11" title="LPSPACE2:func.11">Lp-Norm</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r0</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 ;<br/>
<a NAME="E7:87"/>
<font color="Maroon" title="c6">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="lpspace2.html#T58" target="_self" title="LPSPACE2:th.58">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:87"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="normsp_0.html#D6" title="NORMSP_0:def.6">NORMSP_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c7">x</font> <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="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> ) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c7">x</font> <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="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> ) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><a NAME="E1:87_1"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#D1" target="_self" title="LPSPACE2:def.1">Def1</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:87_1_1"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c7">x</font> <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="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c10">f</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="E3:87_1_1"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp;  ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#D12" target="_self" title="LPSPACE2:def.12">Def12</a></span>;<br/><a NAME="E4:87_1_1"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c10">f</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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T51" target="_self" title="LPSPACE2:th.51">Th52</a>, <a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:87_1_1"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
( <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <font color="Maroon" title="c11">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="lpspace2.html#K11" title="LPSPACE2:func.11">Lp-Norm</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="lpspace2.html#T49" target="_self" title="LPSPACE2:th.49">Th49</a></span>;<br/><a NAME="E7:87_1_1"/>
<font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_1"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E6:87_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E8:87_1_1"/><span class="kw">then </span>
 <a href="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c10">f</font>,<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="#E4:87_1_1"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E6:87_1_1"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="lpspace2.html#T57" target="_self" title="LPSPACE2:th.57">Th58</a>, <a class="ref" href="lpspace2.html#T42" target="_self" title="LPSPACE2:th.42">Th42</a></span>;<br/><a NAME="E9:87_1_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</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="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c10">f</font>,<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#D11" target="_self" title="LPSPACE2:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E10:87_1_1"/>
<font color="Maroon" title="c7">x</font> <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="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="lpspace2.html#T55" target="_self" title="LPSPACE2:th.55">Th56</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/></div>
<span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c10">f</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="E4:87_1"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp;  ex <font color="Olive" title="b1">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#D12" target="_self" title="LPSPACE2:def.12">Def12</a></span>;<br/><a NAME="E5:87_1"/><span class="lab"><font color="Green" title="E70">A8</font></span>: 
( <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c10">f</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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T51" target="_self" title="LPSPACE2:th.51">Th52</a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">g</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="E7:87_1"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
( <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> &amp;  ex <font color="Olive" title="b1">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r2</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#D12" target="_self" title="LPSPACE2:def.12">Def12</a></span>;<br/><a NAME="E8:87_1"/><span class="lab"><font color="Green" title="E72">A10</font></span>: 
( <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c11">g</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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T51" target="_self" title="LPSPACE2:th.51">Th52</a>, <a class="txt" href="#E7:87_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">s1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E10:87_1"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
( <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/><a NAME="E11:87_1"/><span class="lab"><font color="Green" title="E74">A12</font></span>: 
( <font color="Maroon" title="c12">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="power.html#D2" title="POWER:def.2">POWER:def 2</a></span>;<br/><a NAME="E12:87_1"/>
( <font color="Maroon" title="c12">s1</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>  implies <font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><span class="kw">hence </span><a NAME="E13:87_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:87_1"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E10:87_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="lpspace2.html#T49" target="_self" title="LPSPACE2:th.49">Th49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c10">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">g</font>;<br/><span class="kw">set </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font>;<br/><a NAME="E14:87_1"/><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<font color="Maroon" title="c12">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E10:87_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="lpspace2.html#T49" target="_self" title="LPSPACE2:th.49">Th49</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">s2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E16:87_1"/><span class="lab"><font color="Green" title="E76">A14</font></span>: 
( <font color="Maroon" title="c15">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s2</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a></span>;<br/><a NAME="E17:87_1"/>
<font color="Maroon" title="c10">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T54" target="_self" title="LPSPACE2:th.54">Th55</a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E7:87_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a></span>;<br/><a NAME="E18:87_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">g</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T53" target="_self" title="LPSPACE2:th.53">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E19:87_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T61" target="_self" title="LPSPACE2:th.61">Th62</a>, <a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E8:87_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E16:87_1"><span class="lab"><font color="Green" title="E76">A14</font></span></a>, <a class="txt" href="#E10:87_1"><span class="lab"><font color="Green" title="E73">A11</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"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">set </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c9">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c10">f</font>;<br/><span class="kw">set </span><font color="Maroon" title="c17">w</font> = <font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font>;<br/><a NAME="E20:87_1"/>
<font color="Maroon" title="c9">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T54" target="_self" title="LPSPACE2:th.54">Th55</a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/><a NAME="E21:87_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T53" target="_self" title="LPSPACE2:th.53">Th54</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E23:87_1"/><span class="lab"><font color="Green" title="E77">A15</font></span>: 
( <font color="Maroon" title="c18">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 ;<br/><span class="lab"><font color="Green" title="E78">A16</font></span>: <font color="Maroon" title="c18">s</font> = 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="lpspace2.html#T18" target="_self" title="LPSPACE2:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c12">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">s1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T5" title="EXTREAL1:th.5">EXTREAL1:5</a></span>
;<br/><a NAME="E25:87_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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#T4" target="_self" title="LPSPACE2:th.4">Th4</a>, <a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87_1"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E23:87_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="txt" href="#E24:87_1"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="ref" href="lpspace2.html#T5" target="_self" title="LPSPACE2:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a>, <a class="ref" href="holder_1.html#T2" title="HOLDER_1:th.2">HOLDER_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">s1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>
;<br/><span class="kw">hence </span><a NAME="E27:87_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:87"/>
(  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K12" title="LPSPACE2:func.12">Lp-Space</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a>, <a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a>, <a class="ref" href="rsspace3.html#T2" title="RSSPACE3:th.2">RSSPACE3:2</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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