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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" 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>  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="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  iff  ex <font color="Olive" title="b2">I</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>  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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="integra3.html#K2" title="INTEGRA3:func.2">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="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  iff  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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="E1:25"/><span class="lab"><font color="Green" title="E17">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  iff  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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="firebrick">::  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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">I</font> =  <a href="integr16.html#K6" title="INTEGR16:func.6">integral</a> <font color="Maroon" title="c2">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/><span class="kw">assume </span><a NAME="E2:25_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="c4">I</font> = <font color="Maroon" title="c3">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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> <font color="Maroon" title="c4">I</font> )</span><br/><span class="kw">thus </span><a NAME="E3:25_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="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  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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> <font color="Maroon" title="c4">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:25_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="integr16.html#T14" target="_self" title="INTEGR16:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_2"/>
 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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">I</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="E3:25_2"/><span class="lab"><font color="Green" title="E18">A3</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="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  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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> <font color="Maroon" title="c3">I</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">Ii</font> =  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">I</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">reconsider </span><font color="Maroon" title="c5">Ic</font> =  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">I</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/><div><span class="lab"><font color="Green" title="E19">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">I</font>;<br/><span class="kw">let </span><font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Si</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</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="c7">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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> (<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>,<font color="Olive" title="b1">Si</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> (<span class="p2">(<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>,<font color="Olive" title="b1">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ii</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">Si</font> be    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</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="c7">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ii</font> ) )</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1 &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">z</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="Olive" title="b1">H</font> &amp; (  for <font color="Olive" title="b3">j</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>   st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">H</font> holds <br/> ex <font color="Olive" title="b4">rji</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>  ex <font color="Olive" title="b5">tji</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="b5">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> $1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">tji</font></span>)</span> &amp; <font color="Olive" title="b4">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">tji</font> &amp; <font color="Olive" title="b2">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b4">rji</font> ) ) );<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">xs</font> = <font color="Maroon" title="c3">I</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/><a NAME="E2:25_2_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 for <font color="Olive" title="b1">k</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>   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#K2" title="NUMBERS:func.2">COMPLEX</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">k</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1"><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#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="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#K2" title="NUMBERS:func.2">COMPLEX</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">k</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">Tk</font> = <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">j</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>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp;  ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">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; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> ) );<br/>
<a NAME="E2:25_2_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">j0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> implies  ex <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_2_1_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c12">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j0</font> as    <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>  ;<br/>
<a NAME="E3:25_2_1_1_1"/>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_1_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></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#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:25_2_1_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<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="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="c7">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="E6:25_2_1_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="binop_2.html#K11" title="BINOP_2:func.11">*</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="c7">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="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">tji</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:25_2_1_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c15">tji</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"><span class="p2">(<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="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="c7">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="E9:25_2_1_1_1"/><span class="lab"><font color="Green" title="E26">A11</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"><span class="p2">(<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="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="c7">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">tji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2_1_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:25_2_1_1_1"/>
<font color="Maroon" title="c15">tji</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> <span class="p2">(<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>)</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="c7">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="#E8:25_2_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">tji</font> = <font color="Maroon" title="c15">tji</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="lab"><font color="Green" title="E27">A12</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="c7">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="c7">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" title="RELAT_1:th.61">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> <span class="p2">(<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>)</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="c7">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" title="COMSEQ_3:def.3">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"><span class="p2">(<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="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="c7">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" title="RELAT_1:th.61">RELAT_1:61</a></span>
;<br/>
<a NAME="E13:25_2_1_1_1"/><span class="kw">then </span>
<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="c7">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="c16">tji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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="#E8:25_2_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">rji</font> = <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="c7">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="c16">tji</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <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="c7">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="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c17">rji</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Maroon" title="c18">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E16:25_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Maroon" title="c18">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_1_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:25_2_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E9:25_2_1_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E12:25_2_1_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c12">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E4:25_2_1_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <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> <font color="Maroon" title="c11">Tk</font></span>)</span> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E2:25_2_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c13">x</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E23">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">jj0</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font> implies  ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">j0</font> = <font color="Maroon" title="c14">jj0</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E2:25_2_1_1_2"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font> <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> <font color="Maroon" title="c11">Tk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:25_2_1_1_2"/>
<font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> )</span><br/><a NAME="E4:25_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">j0</font>,<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j0</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E2:25_2_1_1_2"><span class="lab"><font color="Green" title="E24">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:25_2_1_1_2"/>
 ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:25_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</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="c11">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:25_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c13">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c10">S</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:25_2_1"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
 for <font color="Olive" title="b1">x</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="#E2:25_2_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>);<br/></span><a NAME="E5:25_2_1"/>
 for <font color="Olive" title="b1">k</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  <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_2"><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#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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">H</font>, <font color="Maroon" title="c13">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:25_2_1_2"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
( <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">H</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="c13">z</font> )
 <span class="kw">and </span><br/><a NAME="E3:25_2_1_2"/><span class="lab"><font color="Green" title="E23">A18</font></span>: 
 for <font color="Olive" title="b1">j</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>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font> holds <br/> ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">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; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; <font color="Maroon" title="c13">z</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> <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="c7">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> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font> implies  ex <font color="Olive" title="b1">rji</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>  st <br/>( <font color="Olive" title="b1">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:25_2_1_2_1"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rji</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>  st <br/>( <font color="Olive" title="b1">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">x</font> as    <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>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c16">rji</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> , <font color="Maroon" title="c17">tji</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> <span class="kw"> such that </span><br/><a NAME="E4:25_2_1_2_1"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<font color="Maroon" title="c17">tji</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="c7">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="c15">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:25_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">tji</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:25_2_1_2_1"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
<font color="Maroon" title="c16">rji</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">tji</font>
 <span class="kw">and </span><br/><a NAME="E7:25_2_1_2_1"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c16">rji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_1_2"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="txt" href="#E1:25_2_1_2_1"><span class="lab"><font color="Green" title="E25">A20</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">rji</font> = <font color="Maroon" title="c16">rji</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E8:25_2_1_2_1"/>
<font color="Maroon" title="c18">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_1_2_1"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E6:25_2_1_2_1"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span></span><br/><span class="kw">thus </span><a NAME="E9:25_2_1_2_1"/>
<font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_1_2_1"><span class="lab"><font color="Green" title="E28">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:25_2_1_2"/>
<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:25_2_1_2"/>
<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">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="#E2:25_2_1_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E4:25_2_1_2"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="ref" href="integr16.html#D1" target="_self" title="INTEGR16:def.1">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">S</font> = <font color="Maroon" title="c10">S</font> as    <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="c7">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D3" target="_self" title="INTEGR16:def.3">Def5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">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="c11">S</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">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="c11">S</font>) as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><span class="kw">assume </span><a NAME="E8:25_2_1"/>
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ii</font> )</span><br/><a NAME="E9:25_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A24</font></span>: 
(  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="c11">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="#E3:25_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">rseqi</font> =  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</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="c11">S</font>)</span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/><a NAME="E11:25_2_1"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
 for <font color="Olive" title="b1">k</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 <br/> ( <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c14">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c9">xs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c14">rseqi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25_2_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="comseq_3.html#T41" title="COMSEQ_3:th.41">COMSEQ_3:41</a>, <a class="ref" href="comseq_3.html#D5" title="COMSEQ_3:def.5">COMSEQ_3:def 5</a></span>;<br/><a NAME="E12:25_2_1"/>
 for <font color="Olive" title="b1">k</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  <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">H</font>, <font color="Maroon" title="c17">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:25_2_1_3"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c16">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E3:25_2_1_3"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E4:25_2_1_3"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">H</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="c17">z</font>
 <span class="kw">and </span><br/><a NAME="E5:25_2_1_3"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
 for <font color="Olive" title="b1">j</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>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">H</font> holds <br/> ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; <font color="Maroon" title="c17">z</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> <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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A30</font></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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">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> <font color="Maroon" title="c16">H</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_1_3"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E4:25_2_1_3"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1: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="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_1_3"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E7:25_2_1_3"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> holds <br/><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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">j0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> implies <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c19">j</font> = <font color="Maroon" title="c18">j0</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:25_2_1_3_2"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integr16.html#K1" title="INTEGR16:func.1">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font></span><br/>

<a NAME="E3:25_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_1_3"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:25_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c16">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_1_3"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>;<br/>
<a NAME="E5:25_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">rji</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> , <font color="Maroon" title="c21">tji</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> <span class="kw"> such that </span><br/><a NAME="E7:25_2_1_3_2"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c21">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:25_2_1_3_2"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:25_2_1_3_2"/><span class="lab"><font color="Green" title="E33">A35</font></span>: 
<font color="Maroon" title="c20">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">tji</font>
 <span class="kw">and </span><br/><a NAME="E10:25_2_1_3_2"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c20">rji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2_1_3"><span class="lab"><font color="Green" title="E27">A29</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A37</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">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> <span class="p2">(<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>)</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">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"><span class="p2">(<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="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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
;<br/>
<a NAME="E12:25_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">tji</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> <span class="p2">(<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>)</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_1_3_2"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E13:25_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A38</font></span>: 
<font color="Maroon" title="c21">tji</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="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E37">A39</font></span>: <span class="p1">(<span class="default"><span class="p2">(<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="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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font> = 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_1_3_2"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E11:25_2_1_3_2"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
 <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="c21">tji</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_2_1_3_2"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c20">rji</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_1_3_2"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E9:25_2_1_3_2"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_1_3_2"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25_2_1_3_2"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="txt" href="#E14:25_2_1_3_2"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="ref" href="integr16.html#T1" target="_self" title="INTEGR16:th.1">Th1</a>, <a class="txt" href="#E3:25_2_1_3"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E10:25_2_1_3_2"><span class="lab"><font color="Green" title="E34">A36</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:25_2_1_3_2"/>
<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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:25_2_1_3"/><span class="lab"><font color="Green" title="E30">A40</font></span>: 
 for <font color="Olive" title="b1">j0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> holds <br/><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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_1_3"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">xseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D5" title="COMSEQ_3:def.5">COMSEQ_3:def 5</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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D4" target="_self" title="INTEGR16:def.4">Def6</a></span>
<br/>.= 
 <a href="integr15.html#K1" title="INTEGR15:func.1">middle_sum</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="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_1_3"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E8:25_2_1_3"><span class="lab"><font color="Green" title="E30">A40</font></span></a>, <a class="ref" href="integr16.html#T7" target="_self" title="INTEGR16:th.7">Th7</a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D4" title="INTEGR15:def.4">INTEGR15:def 4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:25_2_1"/>
(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ii</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25_2_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:25_2"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</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="#E1:25"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="integr16.html#T13" target="_self" title="INTEGR16:th.13">Th13</a></span>;<br/><a NAME="E8:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A41</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="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="integr15.html#T10" title="INTEGR15:th.10">INTEGR15:10</a></span>;<br/><div><span class="lab"><font color="Green" title="E21">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">I</font>;<br/><span class="kw">let </span><font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Si</font> being    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</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="c7">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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> (<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>,<font color="Olive" title="b1">Si</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> (<span class="p2">(<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>,<font color="Olive" title="b1">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ic</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">Si</font> be    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</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="c7">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ic</font> ) )</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1 &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">z</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="Olive" title="b1">H</font> &amp; (  for <font color="Olive" title="b3">j</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>   st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">H</font> holds <br/> ex <font color="Olive" title="b4">rji</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>  ex <font color="Olive" title="b5">tji</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="b5">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> $1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">tji</font></span>)</span> &amp; <font color="Olive" title="b4">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">tji</font> &amp; <font color="Olive" title="b2">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1</span>)</span>,<font color="Olive" title="b3">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b4">rji</font> ) ) );<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">xs</font> = <font color="Maroon" title="c3">I</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/><a NAME="E2:25_2_2"/><span class="lab"><font color="Green" title="E22">A43</font></span>: 
 for <font color="Olive" title="b1">k</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>   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#K2" title="NUMBERS:func.2">COMPLEX</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">k</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1"><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#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="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#K2" title="NUMBERS:func.2">COMPLEX</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">k</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">Tk</font> = <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c10">k</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">j</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>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp;  ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">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; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> ) );<br/>
<a NAME="E2:25_2_2_1"/><span class="lab"><font color="Green" title="E23">A44</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">j0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> implies  ex <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_2_2_1_1"/><span class="lab"><font color="Green" title="E24">A45</font></span>: 
<font color="Maroon" title="c12">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j0</font> as    <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>  ;<br/>
<a NAME="E3:25_2_2_1_1"/>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_2_1_1"><span class="lab"><font color="Green" title="E24">A45</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></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#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:25_2_2_1_1"/><span class="lab"><font color="Green" title="E25">A46</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<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="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="c7">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="E6:25_2_2_1_1"/><span class="lab"><font color="Green" title="E26">A47</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="binop_2.html#K11" title="BINOP_2:func.11">*</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="c7">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="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">tji</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:25_2_2_1_1"/><span class="lab"><font color="Green" title="E27">A48</font></span>: 
<font color="Maroon" title="c15">tji</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"><span class="p2">(<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="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="c7">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="E9:25_2_2_1_1"/><span class="lab"><font color="Green" title="E28">A49</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"><span class="p2">(<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="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="c7">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">tji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2_2_1_1"><span class="lab"><font color="Green" title="E25">A46</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:25_2_2_1_1"/>
<font color="Maroon" title="c15">tji</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> <span class="p2">(<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>)</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="c7">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="#E8:25_2_2_1_1"><span class="lab"><font color="Green" title="E27">A48</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">tji</font> = <font color="Maroon" title="c15">tji</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="lab"><font color="Green" title="E29">A50</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="c7">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="c7">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" title="RELAT_1:th.61">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> <span class="p2">(<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>)</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="c7">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#D4" title="COMSEQ_3:def.4">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"><span class="p2">(<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="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="c7">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" title="RELAT_1:th.61">RELAT_1:61</a></span>
;<br/>
<a NAME="E13:25_2_2_1_1"/><span class="kw">then </span>
<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="c7">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="c16">tji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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="#E8:25_2_2_1_1"><span class="lab"><font color="Green" title="E27">A48</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">rji</font> = <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="c7">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="c16">tji</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <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="c7">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="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c17">rji</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Maroon" title="c18">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E16:25_2_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j0</font>,<font color="Maroon" title="c18">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_2_1_1"><span class="lab"><font color="Green" title="E26">A47</font></span></a>, <a class="txt" href="#E8:25_2_2_1_1"><span class="lab"><font color="Green" title="E27">A48</font></span></a>, <a class="txt" href="#E9:25_2_2_1_1"><span class="lab"><font color="Green" title="E28">A49</font></span></a>, <a class="txt" href="#E12:25_2_2_1_1"><span class="lab"><font color="Green" title="E29">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c12">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E4:25_2_2_1"/><span class="lab"><font color="Green" title="E24">A51</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <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> <font color="Maroon" title="c11">Tk</font></span>)</span> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">Tk</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E2:25_2_2_1"><span class="lab"><font color="Green" title="E23">A44</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c13">x</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E25">A52</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">jj0</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font> implies  ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">j0</font> = <font color="Maroon" title="c14">jj0</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E2:25_2_2_1_2"/><span class="lab"><font color="Green" title="E26">A53</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font> <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> <font color="Maroon" title="c11">Tk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:25_2_2_1_2"/>
<font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Tk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> )</span><br/><a NAME="E4:25_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">j0</font>,<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j0</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_2_1"><span class="lab"><font color="Green" title="E24">A51</font></span></a>, <a class="txt" href="#E2:25_2_2_1_2"><span class="lab"><font color="Green" title="E26">A53</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:25_2_2_1_2"/>
 ex <font color="Olive" title="b1">rji</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>  ex <font color="Olive" title="b2">tji</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">tji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">tji</font></span>)</span> &amp; <font color="Olive" title="b1">rji</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">tji</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">jj0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c14">jj0</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b1">rji</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:25_2_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</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="c11">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_2_1"><span class="lab"><font color="Green" title="E24">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:25_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c13">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_2_1"><span class="lab"><font color="Green" title="E25">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c10">S</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:25_2_2"/><span class="lab"><font color="Green" title="E23">A54</font></span>: 
 for <font color="Olive" title="b1">x</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="#E2:25_2_2"><span class="lab"><font color="Green" title="E22">A43</font></span></a>);<br/></span><a NAME="E5:25_2_2"/>
 for <font color="Olive" title="b1">k</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  <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_2"><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#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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">H</font>, <font color="Maroon" title="c13">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:25_2_2_2"/><span class="lab"><font color="Green" title="E24">A55</font></span>: 
( <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">H</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="c13">z</font> )
 <span class="kw">and </span><br/><a NAME="E3:25_2_2_2"/><span class="lab"><font color="Green" title="E25">A56</font></span>: 
 for <font color="Olive" title="b1">j</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>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font> holds <br/> ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">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; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; <font color="Maroon" title="c13">z</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> <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="c7">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> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_2"><span class="lab"><font color="Green" title="E23">A54</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font> implies  ex <font color="Olive" title="b1">rji</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>  st <br/>( <font color="Olive" title="b1">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:25_2_2_2_1"/><span class="lab"><font color="Green" title="E27">A58</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rji</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>  st <br/>( <font color="Olive" title="b1">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">x</font> as    <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>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c16">rji</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> , <font color="Maroon" title="c17">tji</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> <span class="kw"> such that </span><br/><a NAME="E4:25_2_2_2_1"/><span class="lab"><font color="Green" title="E28">A59</font></span>: 
<font color="Maroon" title="c17">tji</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="c7">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="c15">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:25_2_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">tji</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:25_2_2_2_1"/><span class="lab"><font color="Green" title="E29">A60</font></span>: 
<font color="Maroon" title="c16">rji</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">tji</font>
 <span class="kw">and </span><br/><a NAME="E7:25_2_2_2_1"/><span class="lab"><font color="Green" title="E30">A61</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">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="c15">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c16">rji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_2_2"><span class="lab"><font color="Green" title="E25">A56</font></span></a>, <a class="txt" href="#E1:25_2_2_2_1"><span class="lab"><font color="Green" title="E27">A58</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">rji</font> = <font color="Maroon" title="c16">rji</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E8:25_2_2_2_1"/>
<font color="Maroon" title="c18">rji</font> <a href="hidden.html#R2" title="HIDDEN: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="c7">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">x</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_2_2_1"><span class="lab"><font color="Green" title="E28">A59</font></span></a>, <a class="txt" href="#E6:25_2_2_2_1"><span class="lab"><font color="Green" title="E29">A60</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span></span><br/><span class="kw">thus </span><a NAME="E9:25_2_2_2_1"/>
<font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">rji</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c7">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">x</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_2_2_1"><span class="lab"><font color="Green" title="E30">A61</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:25_2_2_2"/>
<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:25_2_2_2"/>
<font color="Maroon" title="c10">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">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="#E2:25_2_2_2"><span class="lab"><font color="Green" title="E24">A55</font></span></a>, <a class="txt" href="#E4:25_2_2_2"><span class="lab"><font color="Green" title="E26">A57</font></span></a>, <a class="ref" href="integr16.html#D1" target="_self" title="INTEGR16:def.1">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">S</font> = <font color="Maroon" title="c10">S</font> as    <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="c7">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D3" target="_self" title="INTEGR16:def.3">Def5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">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="c11">S</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">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="c11">S</font>) as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><span class="kw">assume </span><a NAME="E8:25_2_2"/>
(  <a href="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="integra3.html#K2" title="INTEGRA3:func.2">delta</a> <font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ic</font> )</span><br/><a NAME="E9:25_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A62</font></span>: 
(  <a href="integr16.html#K5" title="INTEGR16:func.5">middle_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="c11">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="#E3:25_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">rseqi</font> =  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</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="c11">S</font>)</span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/><a NAME="E11:25_2_2"/><span class="lab"><font color="Green" title="E25">A63</font></span>: 
 for <font color="Olive" title="b1">k</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 <br/> ( <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c14">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c9">xs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c14">rseqi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25_2_2"><span class="lab"><font color="Green" title="E24">A62</font></span></a>, <a class="ref" href="comseq_3.html#T41" title="COMSEQ_3:th.41">COMSEQ_3:41</a>, <a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</a></span>;<br/><a NAME="E12:25_2_2"/>
 for <font color="Olive" title="b1">k</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  <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">H</font>, <font color="Maroon" title="c17">z</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:25_2_2_3"/><span class="lab"><font color="Green" title="E26">A64</font></span>: 
<font color="Maroon" title="c16">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E3:25_2_2_3"/><span class="lab"><font color="Green" title="E27">A65</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E4:25_2_2_3"/><span class="lab"><font color="Green" title="E28">A66</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">H</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="c17">z</font>
 <span class="kw">and </span><br/><a NAME="E5:25_2_2_3"/><span class="lab"><font color="Green" title="E29">A67</font></span>: 
 for <font color="Olive" title="b1">j</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>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">H</font> holds <br/> ex <font color="Olive" title="b2">rji</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>  ex <font color="Olive" title="b3">tji</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="b3">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">tji</font></span>)</span> &amp; <font color="Olive" title="b2">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">tji</font> &amp; <font color="Maroon" title="c17">z</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> <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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Olive" title="b2">rji</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_2_2"><span class="lab"><font color="Green" title="E23">A54</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E30">A68</font></span>:  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">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> <font color="Maroon" title="c16">H</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_2_3"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E4:25_2_2_3"><span class="lab"><font color="Green" title="E28">A66</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1: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="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_2_3"><span class="lab"><font color="Green" title="E26">A64</font></span></a>, <a class="ref" href="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E7:25_2_2_3"/><span class="lab"><font color="Green" title="E31">A69</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> holds <br/><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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">j0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> implies <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c19">j</font> = <font color="Maroon" title="c18">j0</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:25_2_2_3_2"/><span class="lab"><font color="Green" title="E32">A70</font></span>: 
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integr16.html#K2" title="INTEGR16:func.2">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font></span><br/>

<a NAME="E3:25_2_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2_2_3"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:25_2_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c16">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_2_3"><span class="lab"><font color="Green" title="E26">A64</font></span></a>, <a class="ref" href="integr15.html#D1" title="INTEGR15:def.1">INTEGR15:def 1</a></span>;<br/>
<a NAME="E5:25_2_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">rji</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> , <font color="Maroon" title="c21">tji</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> <span class="kw"> such that </span><br/><a NAME="E7:25_2_2_3_2"/><span class="lab"><font color="Green" title="E33">A71</font></span>: 
<font color="Maroon" title="c21">tji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:25_2_2_3_2"/><span class="lab"><font color="Green" title="E34">A72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:25_2_2_3_2"/><span class="lab"><font color="Green" title="E35">A73</font></span>: 
<font color="Maroon" title="c20">rji</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">tji</font>
 <span class="kw">and </span><br/><a NAME="E10:25_2_2_3_2"/><span class="lab"><font color="Green" title="E36">A74</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c20">rji</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2_2_3"><span class="lab"><font color="Green" title="E29">A67</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E37">A75</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">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> <span class="p2">(<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>)</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">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"><span class="p2">(<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="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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
;<br/>
<a NAME="E12:25_2_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">tji</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> <span class="p2">(<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>)</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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_2_3_2"><span class="lab"><font color="Green" title="E33">A71</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E13:25_2_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A76</font></span>: 
<font color="Maroon" title="c21">tji</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="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E39">A77</font></span>: <span class="p1">(<span class="default"><span class="p2">(<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="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="c7">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c19">j</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font> = 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">tji</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_2_3_2"><span class="lab"><font color="Green" title="E33">A71</font></span></a>, <a class="txt" href="#E11:25_2_2_3_2"><span class="lab"><font color="Green" title="E37">A75</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
 <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="c21">tji</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_2_2_3_2"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c20">rji</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_2_3_2"><span class="lab"><font color="Green" title="E33">A71</font></span></a>, <a class="txt" href="#E9:25_2_2_3_2"><span class="lab"><font color="Green" title="E35">A73</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2_2_3_2"><span class="lab"><font color="Green" title="E32">A70</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25_2_2_3_2"><span class="lab"><font color="Green" title="E34">A72</font></span></a>, <a class="txt" href="#E14:25_2_2_3_2"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="txt" href="#E3:25_2_2_3"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E10:25_2_2_3_2"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="ref" href="integr16.html#T1" target="_self" title="INTEGR16:th.1">Th1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:25_2_2_3_2"/>
<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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">j0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:25_2_2_3"/><span class="lab"><font color="Green" title="E32">A78</font></span>: 
 for <font color="Olive" title="b1">j0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> holds <br/><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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2_2_3"><span class="lab"><font color="Green" title="E31">A69</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c14">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">xseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</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="c11">S</font> <a href="integr16.html#K4" title="INTEGR16:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#D4" target="_self" title="INTEGR16:def.4">Def6</a></span>
<br/>.= 
 <a href="integr15.html#K1" title="INTEGR15:func.1">middle_sum</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="p1">(<span class="default"><font color="Maroon" title="c8">Si</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c15">k</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25_2_2_3"><span class="lab"><font color="Green" title="E32">A78</font></span></a>, <a class="txt" href="#E6:25_2_2_3"><span class="lab"><font color="Green" title="E30">A68</font></span></a>, <a class="ref" href="integr16.html#T8" target="_self" title="INTEGR16:th.8">Th8</a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D4" title="INTEGR15:def.4">INTEGR15:def 4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:25_2_2"/>
(  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</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>,<font color="Maroon" title="c8">Si</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> (<span class="p2">(<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>,<font color="Maroon" title="c8">Si</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ic</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25_2_2"><span class="lab"><font color="Green" title="E25">A63</font></span></a>, <a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:25_2"/>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</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="#E1:25"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="integr16.html#T13" target="_self" title="INTEGR16:th.13">Th13</a></span>;<br/><a NAME="E11:25_2"/><span class="kw">then </span>
 <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="#E9:25_2"><span class="lab"><font color="Green" title="E21">A42</font></span></a>, <a class="ref" href="integr15.html#T10" title="INTEGR15:th.10">INTEGR15:10</a></span>;<br/><span class="kw">hence </span><a NAME="E12:25_2"/>
<font color="Maroon" title="c2">f</font> is  <a href="integr16.html#V1" title="INTEGR16:attr.1">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25_2"><span class="lab"><font color="Green" title="E20">A41</font></span></a>, <a class="ref" href="integr16.html#D5" target="_self" title="INTEGR16:def.5">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:25"/>
(  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#K2" title="NUMBERS:func.2">COMPLEX</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="integr16.html#M2" title="INTEGR16:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">T</font>  st  <a href="integra3.html#K2" title="INTEGRA3:func.2">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="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#K6" title="NUMBERS:func.6">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="b3">S</font>) is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">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="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="integr16.html#V1" title="INTEGR16:attr.1">integrable</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
