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<span class="kw">let </span><font color="Maroon" title="c1">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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">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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">y</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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b2">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font> st <br/>( <font color="Olive" title="b3">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">D</font> &amp; <font color="Olive" title="b3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">A</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">y</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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">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"> (  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">y</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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> implies  ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">y</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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> or  ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:17"/>
<font color="Maroon" title="c1">y</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#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">D3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c4">D3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:17"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">D3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">D3</font> = <font color="Maroon" title="c4">D3</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D3" title="INTEGRA1:def.3">INTEGRA1:def 3</a></span>;<br/>
<a NAME="E7:17"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E8:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
1 <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="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></span>;<br/>
<a NAME="E9:17"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</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="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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">  ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )</span><br/><div class="add"><a NAME="E2:17_1_1_1"/>
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></span>;<br/><a NAME="E3:17_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/><a NAME="E4:17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_1_1_1"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E20">A3</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_2"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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">  ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )</span><br/><div class="add"><a NAME="E2:17_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:17_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <font color="Maroon" title="c5">D3</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E2:17_1_1_2_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></span>;<br/>
<a NAME="E3:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A10</font></span>: <font color="Maroon" title="c1">y</font> = 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></span>
;<br/>
<a NAME="E5:17_1_1_2_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E6:17_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</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="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E22">A5</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="E7:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E5:17_1_1_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E3:17_1_1_2_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a>, <a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c5">D3</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 as   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T36" title="INTEGRA1:th.36">INTEGRA1:36</a></span>;<br/>
<a NAME="E9:17_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<a NAME="E10:17_1_1_2_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_2_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a></span>;<br/>
<a NAME="E11:17_1_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E9:17_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> = <font color="Maroon" title="c6">D</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E13:17_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</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="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E14:17_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>;<br/>
<a NAME="E15:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E13:17_1_1_2_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<a NAME="E16:17_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E17:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E18:17_1_1_2_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E19:17_1_1_2_1"/><span class="kw">then </span>
2 <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="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E20:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E21:17_1_1_2_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E22:17_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></span>;<br/>
<a NAME="E23:17_1_1_2_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:17_1_1_2_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">D</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17_1_1_2_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E25:17_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E26:17_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E27:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E28:17_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E39">A23</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1_1_2_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:17_1_1_2_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

;<br/>
<a NAME="E30:17_1_1_2_1"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1_1_2_1_3"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E2:17_1_1_2_1_3"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E3:17_1_1_2_1_3"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E4:17_1_1_2_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:17_1_1_2_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E2:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E5:17_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:17_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">D3</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="E7:17_1_1_2_1_3"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:17_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">D3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E44">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 NAME="E2:17_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E3:17_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A33</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/>
<a NAME="E4:17_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c9">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="c8">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E29:17_1_1_2_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:17_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<a NAME="E6:17_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_1_2_1_3_1_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span><br/><div class="add"><a NAME="E2:17_1_1_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A38</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E3:17_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E55">A39</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E4:17_1_1_2_1_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E3:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E6:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="txt" href="#E5:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_1_1_2_1_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E2:17_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span><br/><div class="add"><a NAME="E2:17_1_1_2_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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="c8">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A41</font></span>: <font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> = 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>

;<br/><a NAME="E4:17_1_1_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E5:17_1_1_2_1_3_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E6:17_1_1_2_1_3_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E6:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="txt" href="#E5:17_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E4:17_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E3:17_1_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:17_1_1_2_1_3_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:17_1_1_2_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:17_1_1_2_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E2:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E9:17_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<a NAME="E10:17_1_1_2_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1_1_2_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E9:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/>
<a NAME="E12:17_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <font color="Maroon" title="c5">D3</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E13:17_1_1_2_1_3"/>
<font color="Maroon" title="c9">i</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="c8">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E29:17_1_1_2_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:17_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c8">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/>
<span class="kw">hence </span><a NAME="E15:17_1_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E7:17_1_1_2_1_3"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</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="E31:17_1_1_2_1"/>
1 <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:17_1_1_2_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E32:17_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:17_1_1_2_1"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>;<br/>
<a NAME="E33:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A45</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17_1_1_2_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">A46</font></span>: 2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
1

;<br/>
<a NAME="E35:17_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c8">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E36:17_1_1_2_1"/><span class="kw">then </span>
1 <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="c8">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E43">A47</font></span>: <font color="Maroon" title="c8">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2

;<br/>
<a NAME="E38:17_1_1_2_1"/>
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D3" title="INTEGRA1:def.3">INTEGRA1:def 3</a></span>;<br/>
<a NAME="E39:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A48</font></span>: 
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></span>;<br/>
<a NAME="E40:17_1_1_2_1"/>
 <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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:17_1_1_2_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E41:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A49</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<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#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></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">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:17_1_1_2_1"><span class="lab"><font color="Green" title="E42">A46</font></span></a>, <a class="ref" href="finseq_6.html#T123" title="FINSEQ_6:th.123">FINSEQ_6:123</a></span>;<br/>
<a NAME="E42:17_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span></span>)</span></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> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E29:17_1_1_2_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E43:17_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A50</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:17_1_1_2_1"><span class="lab"><font color="Green" title="E45">A49</font></span></a>, <a class="txt" href="#E30:17_1_1_2_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<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="Maroon" title="c5">D3</font>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D3</font>,1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c1">y</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_2_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E33:17_1_1_2_1"><span class="lab"><font color="Green" title="E41">A45</font></span></a>, <a class="txt" href="#E1:17_1_1_2_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E43:17_1_1_2_1"><span class="lab"><font color="Green" title="E46">A50</font></span></a>, <a class="ref" href="rvsum_1.html#T106" title="RVSUM_1:th.106">RVSUM_1:106</a></span>
<br/>.= 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></span>
;<br/>
<a NAME="E46:17_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:17_1_1_2_1"><span class="lab"><font color="Green" title="E44">A48</font></span></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:17_1_1_2_1"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E39:17_1_1_2_1"><span class="lab"><font color="Green" title="E44">A48</font></span></a>, <a class="txt" href="#E37:17_1_1_2_1"><span class="lab"><font color="Green" title="E43">A47</font></span></a>, <a class="txt" href="#E20:17_1_1_2_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:17_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:17"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <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"> verum</span><br/>


</div>
