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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><br/>  for <font color="Olive" title="b2">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">S</font> being    <a href="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">T</font>  st <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b2">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">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">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">S</font> being    <a href="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font>  st <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b1">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</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">S</font> being    <a href="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font>  st <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be    <a href="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</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="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  &amp;  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies (  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">seq</font> =  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">xs</font> =  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E3:23"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
(  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="integr16.html#T13" target="_self" onmouseover="rs('integr16/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
(  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="integr16.html#T13" target="_self" onmouseover="rs('integr16/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">xseq</font> =  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>) as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>
<a NAME="E6:23"/>
 ex <font color="Olive" title="b1">rseqi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/> for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b1">rseqi</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">pjinf</font> =  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> $1</span>)</span>;<br/>
<a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:23_1_1"/>
 <a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c9">x</font></span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:23_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">F</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:23_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>);<br/></span>
<a NAME="E5:23_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds   <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</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>
<a NAME="E6:23_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b1">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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"> <font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c10">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">Tk</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">Fk</font> = <font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c10">k</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E2:23_1_3"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Fk</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D1" target="_self" onmouseover="rs('integr16/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<a NAME="E4:23_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A11</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_3_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">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> implies  ex <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:23_1_3_2"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</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">  ex <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">r</font> being    <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"> such that </span><br/><a NAME="E3:23_1_3_2"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:23_1_3_2"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D1" target="_self" onmouseover="rs('integr16/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> =  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c14">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">v</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="c16">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c17">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:23_1_3_2"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c17">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_3_2"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_3_2"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E9:23_1_3_2"/>
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_3_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/><a NAME="E10:23_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">t</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:23_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c17">t</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> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>;<br/><span class="lab"><font color="Green" title="E33">A18</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">pjinf</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
;<br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c14">r</font> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_3_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E8:23_1_3_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_1_3_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_3_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E12:23_1_3_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:23_1_3_2"/>
<font color="Maroon" title="c16">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_3_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E12:23_1_3_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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"> <font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c12">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_3"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E4:23_1_3"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E1:23_1_3_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c16">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c13">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_3_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="integr16.html#T1" target="_self" onmouseover="rs('integr16/T1')" onmouseout="rh()">Th1</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>
<a NAME="E6:23_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_3"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23_1_3"/>
<font color="Maroon" title="c9">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c10">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_3"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="integr15.html#D1" onmouseover="rs('integr15/D1')" onmouseout="rh()">INTEGR15:def 1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">pjis</font> = <font color="Maroon" title="c9">F</font> as    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c3">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D3" onmouseover="rs('integr15/D3')" onmouseout="rh()">INTEGR15:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">rseqi</font> =  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c10">pjis</font>) as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E9:23_1"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c11">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <font color="Maroon" title="c11">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">kk</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:23_1_4"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
 <a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c13">kk</font></span>)</span> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c13">kk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#T4" target="_self" onmouseover="rs('integr16/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c11">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
 <a href="integr15.html#K1" title="INTEGR15:func.1">middle_sum</a> (<font color="Maroon" title="c8">pjinf</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">pjis</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c13">kk</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D4" onmouseover="rs('integr15/D4')" onmouseout="rh()">INTEGR15:def 4</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K3" title="INTEGR16:func.3">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c13">kk</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E2:23_1_4"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="integr16.html#T7" target="_self" onmouseover="rs('integr16/T7')" onmouseout="rh()">Th7</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D4" target="_self" onmouseover="rs('integr16/D4')" onmouseout="rh()">Def4</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">take </span>
<font color="Maroon" title="c11">rseqi</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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c11">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c11">rseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c11">rseqi</font> )</span><br/>

<a NAME="E10:23_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<font color="Maroon" title="c8">pjinf</font>,<font color="Maroon" title="c10">pjis</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c8">pjinf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="integr15.html#T9" onmouseover="rs('integr15/T9')" onmouseout="rh()">INTEGR15:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:23_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c11">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c11">rseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c11">rseqi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:23_1"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="complex1.html#T12" onmouseover="rs('complex1/T12')" onmouseout="rh()">COMPLEX1:12</a>, <a class="ref" href="integr15.html#T9" onmouseover="rs('integr15/T9')" onmouseout="rh()">INTEGR15:9</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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">rseqi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E8:23"/><span class="lab"><font color="Green" title="E20">A21</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c8">rseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c8">rseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c8">rseqi</font> )
 ;<br/>
<a NAME="E9:23"/>
 ex <font color="Olive" title="b1">iseqi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/> for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> &amp; <font color="Olive" title="b1">iseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b1">iseqi</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c9">pjinf</font> =  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> $1</span>)</span>;<br/>
<a NAME="E2:23_2"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:23_2_1"/>
 <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:23_2_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</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"> 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="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:23_2"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:23_2"><span class="lab"><font color="Green" title="E21">A22</font></span></a>);<br/></span>
<a NAME="E5:23_2"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds   <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span></span><br/>

 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:23_2_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:23_2"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b1">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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"> <font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">Tk</font> = <font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Fk</font> = <font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c11">k</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E3:23_2_3"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
<font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_2"><span class="lab"><font color="Green" title="E22">A23</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E25">A26</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">Fk</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_2"><span class="lab"><font color="Green" title="E23">A24</font></span></a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">Tk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D1" target="_self" onmouseover="rs('integr16/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<a NAME="E5:23_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A27</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A28</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2_3_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">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Tk</font> holds <br/> ex <font color="Olive" title="b2">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon" title="c14">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Tk</font> implies  ex <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:23_2_3_2"/><span class="lab"><font color="Green" title="E28">A29</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">Tk</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">  ex <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">r</font> being    <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"> such that </span><br/><a NAME="E3:23_2_3_2"/><span class="lab"><font color="Green" title="E29">A30</font></span>: 
<font color="Maroon" title="c15">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:23_2_3_2"/><span class="lab"><font color="Green" title="E30">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D1" target="_self" onmouseover="rs('integr16/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">v</font> =  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c17">v</font> = <font color="Maroon" title="c16">v</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="c17">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c18">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:23_2_3_2"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
<font color="Maroon" title="c18">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:23_2_3_2"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_2_3_2"><span class="lab"><font color="Green" title="E29">A30</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E9:23_2_3_2"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_2_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/><a NAME="E10:23_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">t</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:23_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon" title="c18">t</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> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E34">A35</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">pjinf</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
;<br/> <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">r</font> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_2_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="txt" href="#E8:23_2_3_2"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_2_3_2"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_2_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="txt" href="#E12:23_2_3_2"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:23_2_3_2"/>
<font color="Maroon" title="c17">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_2_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="txt" href="#E12:23_2_3_2"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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"> <font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c13">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_2_3"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="txt" href="#E5:23_2_3"><span class="lab"><font color="Green" title="E26">A27</font></span></a>, <a class="txt" href="#E1:23_2_3_2"><span class="lab"><font color="Green" title="E28">A29</font></span></a>, <a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c17">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span>,<font color="Maroon" title="c14">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_2_3_2"><span class="lab"><font color="Green" title="E30">A31</font></span></a>, <a class="ref" href="integr16.html#T1" target="_self" onmouseover="rs('integr16/T1')" onmouseout="rh()">Th1</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>
<a NAME="E7:23_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_2_3"><span class="lab"><font color="Green" title="E25">A26</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:23_2_3"/>
<font color="Maroon" title="c10">F</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_2_3"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="ref" href="integr15.html#D1" onmouseover="rs('integr15/D1')" onmouseout="rh()">INTEGR15:def 1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">pjis</font> = <font color="Maroon" title="c10">F</font> as    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c3">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D3" onmouseover="rs('integr15/D3')" onmouseout="rh()">INTEGR15:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">iseqi</font> =  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c11">pjis</font>) as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E9:23_2"/><span class="lab"><font color="Green" title="E24">A36</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c12">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <font color="Maroon" title="c12">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">kk</font> = <font color="Maroon" title="c13">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:23_2_4"/><span class="lab"><font color="Green" title="E25">A37</font></span>: 
 <a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c14">kk</font></span>)</span> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c14">kk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#T4" target="_self" onmouseover="rs('integr16/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c12">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> = 
 <a href="integr15.html#K1" title="INTEGR15:func.1">middle_sum</a> (<font color="Maroon" title="c9">pjinf</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">pjis</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c14">kk</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D4" onmouseover="rs('integr15/D4')" onmouseout="rh()">INTEGR15:def 4</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K3" title="INTEGR16:func.3">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c14">kk</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_2"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E2:23_2_4"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="ref" href="integr16.html#T8" target="_self" onmouseover="rs('integr16/T8')" onmouseout="rh()">Th8</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D4" target="_self" onmouseover="rs('integr16/D4')" onmouseout="rh()">Def4</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">take </span>
<font color="Maroon" title="c12">iseqi</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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c12">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c12">iseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c12">iseqi</font> )</span><br/>

<a NAME="E10:23_2"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<font color="Maroon" title="c9">pjinf</font>,<font color="Maroon" title="c11">pjis</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c9">pjinf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="integr15.html#T9" onmouseover="rs('integr15/T9')" onmouseout="rh()">INTEGR15:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:23_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c12">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c12">iseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c12">iseqi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E9:23_2"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="ref" href="complex1.html#T12" onmouseover="rs('complex1/T12')" onmouseout="rh()">COMPLEX1:12</a>, <a class="ref" href="integr15.html#T9" onmouseover="rs('integr15/T9')" onmouseout="rh()">INTEGR15:9</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">then </span><span class="kw">consider </span><font color="Maroon" title="c9">iseqi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E11:23"/><span class="lab"><font color="Green" title="E21">A38</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c9">iseqi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c9">iseqi</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c9">iseqi</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E12:23"/>
 <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>) is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E21">A38</font></span></a>, <a class="ref" href="comseq_3.html#T38" onmouseover="rs('comseq_3/T38')" onmouseout="rh()">COMSEQ_3:38</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">  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">thus </span> <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p1">(<span class="default"><a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">S</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c8">rseqi</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c9">iseqi</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E21">A38</font></span></a>, <a class="ref" href="comseq_3.html#T39" onmouseover="rs('comseq_3/T39')" onmouseout="rh()">COMSEQ_3:39</a></span>
<br/>.= 
 <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E21">A38</font></span></a>, <a class="ref" href="complex1.html#T13" onmouseover="rs('complex1/T13')" onmouseout="rh()">COMPLEX1:13</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>
