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<span class="kw">let </span><font color="Maroon" title="c1">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <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/> for <font color="Olive" title="b2">D1</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon" title="c1">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="b2">D1</font> holds <br/> <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="Olive" title="b2">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Olive" title="b2">D1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">D1</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c1">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">D1</font> holds <br/> <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="Olive" title="b1">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Olive" title="b1">D1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D1</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">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">D1</font> implies  <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">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c1">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">D1</font>
 ; <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#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">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span><br/>

<a NAME="E2:13"/><span class="kw">then </span>
<font color="Maroon" title="c1">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="c3">D1</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="E3:13"/><span class="kw">then </span>
<font color="Maroon" title="c1">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> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></span>;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
<font color="Maroon" title="c1">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#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</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="E5:13"/><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"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">j</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#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:13"/><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"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2: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#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E7:13"/><span class="kw">then </span>
 <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">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2: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#K13" title="INTEGRA1:func.13">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:13"/>
 <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">D1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</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/>


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