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<span class="kw">let </span><font color="Maroon" title="c5">N1</font>, <font color="Maroon" title="c6">N2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>  ex <font color="Olive" title="b2">f</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> st <br/>( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">f</font></span>)</span>) ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>  ex <font color="Olive" title="b2">f</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> st <br/>( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">f</font></span>)</span>) ) ) implies <font color="Maroon" title="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:78_1_1"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
( (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>  ex <font color="Olive" title="b2">f</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> st <br/>( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">f</font></span>)</span>) ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>  ex <font color="Olive" title="b2">g</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> st <br/>( <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">g</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="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1_1_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">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span> holds  <font color="Maroon" title="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c4">M</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="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E1:78_1_1_1"/>
(  ex <font color="Olive" title="b1">f</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> st <br/>( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">f</font></span>)</span>) ) &amp;  ex <font color="Olive" title="b1">g</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> st <br/>( <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</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="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">g</font></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:78_1_1_1"/>
<font color="Maroon" title="c5">N1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T42" target="_self" onmouseover="rs('lpspacc1/T42')" onmouseout="rh()">Th42</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="E3:78_1_1"/>
<font color="Maroon" title="c5">N1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</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/>


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