<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Olive" title="b3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  holds <br/> <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">g</font></span>)</span>,<font color="Olive" title="b1">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">D</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Olive" title="b2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  holds <br/> <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>  implies  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 <span class="kw">and </span><br/><a NAME="E2:84"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">H</font> =  <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">G</font> =  <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">F</font> =  <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E3:84"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span> <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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></span>;<br/>
<a NAME="E4:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<a NAME="E5:84"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></span>;<br/>
<a NAME="E6:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A4</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<a NAME="E7:84"/><span class="lab"><font color="Green" title="E68">A5</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="c2">D</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></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="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">j</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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="c2">D</font></span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_1"/>
<font color="Maroon" title="c8">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="c2">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span><br/>

<a NAME="E2:84_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="c2">D</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/>
<a NAME="E3:84_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="integra1.html#T55" target="_self" title="INTEGRA1:th.55">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:84_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:84"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></span>;<br/>
<a NAME="E9:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A6</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<a NAME="E10:84"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="finseq_2.html#T140" title="FINSEQ_2:th.140">FINSEQ_2:140</a></span>;<br/>
<a NAME="E11:84"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:84"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="txt" href="#E7:84"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="rvsum_1.html#T112" title="RVSUM_1:th.112">RVSUM_1:112</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:84"/>
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="rvsum_1.html#T119" title="RVSUM_1:th.119">RVSUM_1:119</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
