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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  iff  ex <font color="Olive" title="b2">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b3">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="b4">S</font> being   <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b3">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I</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#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies ( <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  iff  ex <font color="Olive" title="b1">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  iff  ex <font color="Olive" title="b1">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> ) implies <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">I</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/>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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c3">I</font> =  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">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> <font color="Maroon" title="c3">I</font> )</span><br/><span class="kw">thus </span><a NAME="E2:20_1"/>
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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">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> <font color="Maroon" title="c3">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="integr15.html#T9" target="_self" title="INTEGR15:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<a NAME="E3:20"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_2"/>
 ex <font color="Olive" title="b1">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">I</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">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> <font color="Maroon" title="c3">I</font> )
 ;<br/><a NAME="E4:20_2"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
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>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font>;<br/>
<div><span class="lab"><font color="Green" title="E23">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E1:20_2_1_1"/>
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integr15.html#T6" target="_self" title="INTEGR15:th.6">Th6</a></span>;<br/><a NAME="E2:20_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T15" title="VALUED_1:th.15">VALUED_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:20_2_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
(  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font></span><br/>

<a NAME="E4:20_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integra4.html#T9" title="INTEGRA4:th.9">INTEGRA4:9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">e1</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font> implies <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">e1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:20_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">S</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font><span class="kw"> such that </span><br/><a NAME="E4:20_2_1_2"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integr15.html#T8" target="_self" title="INTEGR15:th.8">Th8</a></span>;<br/><div><span class="lab"><font color="Green" title="E28">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">e</font></span><br/><a NAME="E1:20_2_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a></span>;<br/><a NAME="E2:20_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:20_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_2_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T15" title="VALUED_1:th.15">VALUED_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:20_2_1_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/><a NAME="E7:20_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E6:20_2_1_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E9:20_2_1_2"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_1_2"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E7:20_2_1_2"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="prepower.html#T3" title="PREPOWER:th.3">PREPOWER:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:20_2_1"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/>
<a NAME="E7:20_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E6:20_2_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>
;<br/>
<a NAME="E9:20_2_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E7:20_2_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="seq_2.html#T31" title="SEQ_2:th.31">SEQ_2:31</a></span>;<br/>
<a NAME="E10:20_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:20_2_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:20_2"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
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>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font>;<br/>
<div><span class="lab"><font color="Green" title="E24">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E1:20_2_2_1"/>
<span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integr15.html#T5" target="_self" title="INTEGR15:th.5">Th5</a></span>;<br/><a NAME="E2:20_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_2_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T15" title="VALUED_1:th.15">VALUED_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:20_2_2"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
(  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span><br/>

<a NAME="E4:20_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A19</font></span>: 
 <a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integra4.html#T8" title="INTEGRA4:th.8">INTEGRA4:8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">e1</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font> implies <font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c6">e1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:20_2_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">S</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font><span class="kw"> such that </span><br/><a NAME="E4:20_2_2_2"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integr15.html#T7" target="_self" title="INTEGR15:th.7">Th7</a></span>;<br/><div><span class="lab"><font color="Green" title="E29">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">e</font></span><br/><a NAME="E1:20_2_2_2_1"/>
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_2_2"><span class="lab"><font color="Green" title="E28">A21</font></span></a></span>;<br/><a NAME="E2:20_2_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_2_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T15" title="VALUED_1:th.15">VALUED_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:20_2_2_2"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
 <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>;<br/><a NAME="E7:20_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A24</font></span>: 
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E6:20_2_2_2"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E9:20_2_2_2"/>
<font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_2_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E7:20_2_2_2"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="prepower.html#T3" title="PREPOWER:th.3">PREPOWER:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:20_2_2"/><span class="lab"><font color="Green" title="E28">A25</font></span>: 
 <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>;<br/>
<a NAME="E7:20_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A26</font></span>: 
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">S</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E6:20_2_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:20_2_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a></span>
;<br/>
<a NAME="E9:20_2_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E7:20_2_2"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="ref" href="seq_2.html#T31" title="SEQ_2:th.31">SEQ_2:31</a></span>;<br/>
<a NAME="E10:20_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2_2"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:20_2_2"/>
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:20_2_3"/><span class="lab"><font color="Green" title="E24">A27</font></span>: 
(  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K5" title="INTEGRA2:func.5">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E1:20_2_3"><span class="lab"><font color="Green" title="E24">A27</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:20_2"/>
<font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integra4.html#T12" title="INTEGRA4:th.12">INTEGRA4:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:20"/>
(  ex <font color="Olive" title="b1">I</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <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="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">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#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> ) implies <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
