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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c3">D2</font>
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/>
<a NAME="E3:47"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</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"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E5:47"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c4">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="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D2</font></span>)</span> &amp;  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E6:47"/><i><font color="Green" title="E32">X</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">D2</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="c3">D2</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c4">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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E5:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a>, <a class="txt" href="integra2.html#E6:47"><i><font color="Green" title="E32">X</font></i></a></i>;<br/>
<a NAME="E8:47"/><b>then </b><i><font color="Green" title="E34">A6</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E5:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:47"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c5">i</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">D1</font> &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="integra2.html#E7:47"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="integra2.html#T37" target="_self" title="INTEGRA2:th.37">Th37</a></i>;<br/>
<a NAME="E11:47"/><i><font color="Green" title="E36">X</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">D1</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">D1</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E12:47"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c5">i</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="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a>, <a class="txt" href="integra2.html#E10:47"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="integra2.html#E11:47"><i><font color="Green" title="E36">X</font></i></a></i>;<br/>
<a NAME="E13:47"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a>, <a class="txt" href="integra2.html#E10:47"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E14:47"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">i</font></span>)</span> <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"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E12:47"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:47"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="integra2.html#E8:47"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="integra2.html#E10:47"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="integra2.html#T39" target="_self" title="INTEGRA2:th.39">Th39</a>, <a class="ref" href="integra2.html#T40" target="_self" title="INTEGRA2:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E16:47"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c2">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/>


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