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<span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">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> holds <br/> (  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> is  <a href="lpspacc1.html#V1" title="LPSPACC1:attr.1">multi-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">S</font> holds <br/> (  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> is  <a href="lpspacc1.html#V1" title="LPSPACC1:attr.1">multi-closed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp;  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> is  <a href="lpspacc1.html#V1" title="LPSPACC1:attr.1">multi-closed</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">W</font> =  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K12" title="LPSPACC1:func.12">CLSp_PFunct</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K12" title="LPSPACC1:func.12">CLSp_PFunct</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> implies <font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E2:39_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) ;<br/><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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E5:39_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">and </span><br/><a NAME="E6:39_1"/>
 ex <font color="Olive" title="b1">ND</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">ND</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ND</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">v1</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E8:39_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c10">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font>
 <span class="kw">and </span><br/><a NAME="E9:39_1"/>
 ex <font color="Olive" title="b1">ND</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">ND</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ND</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c10">u1</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/><a NAME="E10:39_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">u1</font></span>)</span> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> holds <br/><font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="lpspacc1.html#T4" target="_self" onmouseover="rs('lpspacc1/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E11:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c10">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:39_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:39"/>
 <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> is  <a href="lpspacc1.html#V1" title="LPSPACC1:attr.1">multi-closed</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">u</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K12" title="LPSPACC1:func.12">CLSp_PFunct</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K12" title="LPSPACC1:func.12">CLSp_PFunct</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> holds <br/><font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K12" title="LPSPACC1:func.12">CLSp_PFunct</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> implies <font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )</span><br/><span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E4:39_2"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c9">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font>
 <span class="kw">and </span><br/><a NAME="E5:39_2"/>
 ex <font color="Olive" title="b1">ND</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Maroon" title="c4">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">ND</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ND</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c9">u1</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">h</font> = <font color="Maroon" title="c8">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>) ;<br/><a NAME="E7:39_2"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c8">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K8" title="LPSPACC1:func.8">multcomplexcpfunc</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">u</font>)
 ;<br/><a NAME="E8:39_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="lpspacc1.html#T7" target="_self" onmouseover="rs('lpspacc1/T7')" onmouseout="rh()">Th7</a></span>;<br/><a NAME="E9:39_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> holds <br/><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:39_2"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="lpspacc1.html#T7" target="_self" onmouseover="rs('lpspacc1/T7')" onmouseout="rh()">Th7</a></span>;<br/><a NAME="E10:39_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c9">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_2"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E11:39_2"/>
<font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E4:39_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:39"/>
 <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> is  <a href="lpspacc1.html#V1" title="LPSPACC1:attr.1">multi-closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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