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<span class="kw">let </span><font color="Maroon" title="c1">x</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">D1</font>, <font color="Olive" title="b3">D2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b4">g</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 <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Olive" title="b2">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Olive" title="b4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/><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="Olive" title="b4">g</font>,<font color="Olive" title="b3">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="Olive" title="b4">g</font>,<font color="Olive" title="b2">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b4">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b4">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Olive" title="b2">D1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Olive" title="b1">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b1">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b2">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Olive" title="b3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/><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="Olive" title="b3">g</font>,<font color="Olive" title="b2">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="Olive" title="b3">g</font>,<font color="Olive" title="b1">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Olive" title="b1">D1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">D2</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</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 <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c4">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/><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="Olive" title="b1">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="Olive" title="b1">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</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"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c4">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c3">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c4">D2</font> or  not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> or  not <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  or <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">j</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c4">D2</font>
 <span class="kw">and </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ; <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="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  or <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span> )</span><br/>

<a NAME="E5:15"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 
<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="E6:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E7:15"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">j1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E9:15"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E10:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c7">j1</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> <font color="Maroon" title="c3">D1</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="E11:15"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E12:15"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c3">D1</font>,1,<font color="Maroon" title="c7">j1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a>, <a class="ref" href="integra1.html#T37" title="INTEGRA1:th.37">INTEGRA1:37</a></span>;<br/>
<a NAME="E13:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E14:15"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E15:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E16:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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="E17:15"/><span class="kw">then </span>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c4">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="integra1.html#T37" title="INTEGRA1:th.37">INTEGRA1:37</a></span>;<br/>
<a NAME="E18:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E19:15"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E20:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 
<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="E21:15"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="integra3.html#E14"><span class="lab"><font color="Green" title="E14">Lm6</font></span></a></span>;<br/>
<a NAME="E22:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></span>;<br/>
<a NAME="E23:15"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font> implies <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span><br/>

<span class="kw">assume </span><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_1"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:15_1_1_1"/>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E36">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="E4:15_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E5:15_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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="c4">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font>
 <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="E7:15_1_1_1"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_1"><span class="lab"><font color="Green" title="E41">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="E9:15_1_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/><a NAME="E10:15_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_1"><span class="lab"><font color="Green" title="E41">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="E11:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>;<br/><a NAME="E12:15_1_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E7:15_1_1_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:15_1_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_1"><span class="lab"><font color="Green" title="E41">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="E14:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E40">A25</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="E15:15_1_1_1"/>
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><span class="kw">hence </span><a NAME="E16:15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E11:15_1_1_1"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E14:15_1_1_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E8:15_1_1_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_2"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:15_1_1_2"/>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E36">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="E4:15_1_1_2"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></span>;<br/><a NAME="E5:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A33</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A34</font></span>: 
<font color="Maroon" title="c8">k</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="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E36">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="E8:15_1_1_2"/>
<font color="Maroon" title="c8">k</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> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A35</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>;<br/><a NAME="E10:15_1_1_2"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</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="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_2"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E11:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A36</font></span>: 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E7:15_1_1_2"><span class="lab"><font color="Green" title="E42">A34</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="E12:15_1_1_2"/><span class="lab"><font color="Green" title="E45">A37</font></span>: 
<font color="Maroon" title="c8">k</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="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E5:15_1_1_2"><span class="lab"><font color="Green" title="E41">A33</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E13:15_1_1_2"/>
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_2"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><span class="kw">hence </span><a NAME="E14:15_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E9:15_1_1_2"><span class="lab"><font color="Green" title="E43">A35</font></span></a>, <a class="txt" href="#E11:15_1_1_2"><span class="lab"><font color="Green" title="E44">A36</font></span></a>, <a class="txt" href="#E12:15_1_1_2"><span class="lab"><font color="Green" title="E45">A37</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E24:15"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</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"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> holds <br/><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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:15_2"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</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="E2:15_2"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></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="E3:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A39</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">k</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="c8">k</font> &amp; <font color="Maroon" title="c8">k</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"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> implies <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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:15_2"/><span class="lab"><font color="Green" title="E38">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">and </span><br/><a NAME="E5:15_2"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
<font color="Maroon" title="c8">k</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"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E7:15_2"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
 <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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>
 
<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="E8:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:15_2"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E9:15_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:15_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E11:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A44</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 
<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="E12:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A45</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E13:15_2"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:15_2"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a></span>;<br/>
<a NAME="E15:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A47</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</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="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/>
<a NAME="E16:15_2"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 
<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="E17:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A48</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E18:15_2"/><span class="lab"><font color="Green" title="E47">A49</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E19:15_2"/><span class="lab"><font color="Green" title="E48">A50</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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> &amp;  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_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="E1:15_2_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">k</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="15_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A51</font></span>: 
<font color="Maroon" title="c9">k</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="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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> &amp;  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:15_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A52</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="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E3:15_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A53</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="c3">D1</font>,<font color="Maroon" title="c9">k</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="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E1:15_2_1_1_1"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E4:15_2_1_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E1:15_2_1_1_1"><span class="lab"><font color="Green" title="E49">A51</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:15_2_1_1_1"/>
(  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> &amp;  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E18:15_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E3:15_2_1_1_1"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E2:15_2_1_1_1"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 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="15_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<font color="Maroon" title="c9">k</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="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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> &amp;  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c9">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E3:15_2_1_1_2"/><span class="lab"><font color="Green" title="E50">A55</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E1:15_2_1_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/><a NAME="E4:15_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A56</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k1</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="E5:15_2_1_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:15_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E7:15_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A57</font></span>: 
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:15_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A58</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E1:15_2_1_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a></span>;<br/><a NAME="E9:15_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A59</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="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></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="#E12:15_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E10:15_2_1_1_2"/><span class="lab"><font color="Green" title="E55">A60</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="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E1:15_2_1_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E11:15_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A61</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="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E1:15_2_1_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E12:15_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A62</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="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E8:15_2_1_1_2"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> = 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c10">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E7:15_2_1_1_2"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E4:15_2_1_1_2"><span class="lab"><font color="Green" title="E51">A56</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:15_2_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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> &amp;  <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="c3">D1</font>,<font color="Maroon" title="c9">k</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> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E11:15_2_1_1_2"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="txt" href="#E10:15_2_1_1_2"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="txt" href="#E3:15_2_1_1_2"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E9:15_2_1_1_2"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="txt" href="#E12:15_2_1_1_2"><span class="lab"><font color="Green" title="E57">A62</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</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="E20:15_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font> <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="c3">D1</font>,<font color="Maroon" title="c9">k</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="c3">D1</font>,<font color="Maroon" title="c9">k</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="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/>
<a NAME="E21:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A63</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font> <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="c4">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/>
<a NAME="E22:15_2"/>
<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E7:15_2"><span class="lab"><font color="Green" title="E40">A42</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="E23:15_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A64</font></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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>
<br/>.= 
<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="c5">g</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="c4">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></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="c4">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E21:15_2"><span class="lab"><font color="Green" title="E49">A63</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
;<br/>
<a NAME="E25:15_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a></span>;<br/>
<a NAME="E26:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A65</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_2"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E41">A43</font></span></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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_2"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></span>
<br/>.= 
<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="c5">g</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="c4">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></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="c4">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:15_2"><span class="lab"><font color="Green" title="E51">A65</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E28:15_2"/>
<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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:15_2"><span class="lab"><font color="Green" title="E50">A64</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>
<a NAME="E25:15"/><span class="lab"><font color="Green" title="E37">A66</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</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="c4">D2</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/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font>) <span class="kw">-&gt; </span>   <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>  =  <a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,$1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c2">A</font>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,$1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $2;<br/>
<a NAME="E26:15"/><span class="lab"><font color="Green" title="E38">A67</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c5">g</font>,<font color="Maroon" title="c3">D1</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="E27:15"/><span class="lab"><font color="Green" title="E39">A68</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>)
 
<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="E28:15"/><span class="lab"><font color="Green" title="E40">A69</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E29:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A70</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> <font color="Maroon" title="c3">D1</font>
 
<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="E30:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A71</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:15"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">assume </span><a NAME="E31:15"/><span class="lab"><font color="Green" title="E43">A72</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span></span><br/>

<a NAME="E32:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E33:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A73</font></span>: 
<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E34:15"/><span class="kw">then </span>
<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>)
 
<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="E35:15"/><span class="kw">then </span>
<font color="Maroon" title="c7">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E36:15"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> = 
 <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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</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"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),1,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c7">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E27:15"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="txt" href="#E33:15"><span class="lab"><font color="Green" title="E44">A73</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:15"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
;<br/>
<a NAME="E38:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A74</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:15"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>;<br/>
<a NAME="E39:15"/><span class="lab"><font color="Green" title="E46">A75</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E40:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A76</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E41:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A77</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E42:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E43:15"/><span class="kw">then </span>
<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E44:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A78</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></span>;<br/>
<a NAME="E45:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A79</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E46:15"/><span class="lab"><font color="Green" title="E51">A80</font></span>: 
<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></span>;<br/>
<a NAME="E47:15"/><span class="lab"><font color="Green" title="E52">A81</font></span>: 
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:15_4"/><span class="lab"><font color="Green" title="E53">A82</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">ID2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">assume </span><a NAME="E1:15_4_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A83</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E3:15_4_1"/><span class="lab"><font color="Green" title="E55">A84</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:15_4_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A85</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:15_4_1"/><span class="lab"><font color="Green" title="E57">A86</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E7:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A87</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</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="E8:15_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_4_1"><span class="lab"><font color="Green" title="E54">A83</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A88</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_4_1"><span class="lab"><font color="Green" title="E56">A85</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A89</font></span>: 
<font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_4_1"><span class="lab"><font color="Green" title="E54">A83</font></span></a>, <a class="txt" href="#E6:15_4_1"><span class="lab"><font color="Green" title="E57">A86</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="E11:15_4_1"/><span class="lab"><font color="Green" title="E61">A90</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></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="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:15_4_1"/><span class="lab"><font color="Green" title="E62">A91</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E13:15_4_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_4_1"><span class="lab"><font color="Green" title="E54">A83</font></span></a>, <a class="txt" href="#E3:15_4_1"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:15_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4_1"><span class="lab"><font color="Green" title="E58">A87</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A92</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_4_1"><span class="lab"><font color="Green" title="E61">A90</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E16:15_4_1"/><span class="lab"><font color="Green" title="E64">A93</font></span>: 
<font color="Maroon" title="c3">D1</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="c3">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E17:15_4_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E18:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A94</font></span>: 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E15:15_4_1"><span class="lab"><font color="Green" title="E63">A92</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="E19:15_4_1"/><span class="lab"><font color="Green" title="E66">A95</font></span>: 
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E3:15_4_1"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="txt" href="#E15:15_4_1"><span class="lab"><font color="Green" title="E63">A92</font></span></a>, <a class="txt" href="#E9:15_4_1"><span class="lab"><font color="Green" title="E59">A88</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="E20:15_4_1"/><span class="lab"><font color="Green" title="E67">A96</font></span>: 
( not <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> &amp; not <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_4_1_1"/><span class="lab"><font color="Green" title="E68">A97</font></span>: 
( <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> or <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_4_1_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="E1:15_4_1_1_1"/>
( <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> or <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_4_1_1"><span class="lab"><font color="Green" title="E68">A97</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_4_1_1_1_1"/>
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:15_4_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A98</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:15_4_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A99</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/><a NAME="E5:15_4_1_1_1_1"/>
<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:15"><span class="lab"><font color="Green" title="E51">A80</font></span></a>, <a class="txt" href="#E18:15_4_1"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E12:15_4_1"><span class="lab"><font color="Green" title="E62">A91</font></span></a>, <a class="txt" href="#E3:15_4_1_1_1_1"><span class="lab"><font color="Green" title="E69">A98</font></span></a>, <a class="txt" href="#E4:15_4_1_1_1_1"><span class="lab"><font color="Green" title="E70">A99</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/><a NAME="E6:15_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A100</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> <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="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E7:15_4_1_1_1_1"/>
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_4_1"><span class="lab"><font color="Green" title="E66">A95</font></span></a>, <a class="txt" href="#E10:15_4_1"><span class="lab"><font color="Green" title="E60">A89</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:15_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E16:15_4_1"><span class="lab"><font color="Green" title="E64">A93</font></span></a>, <a class="txt" href="#E3:15_4_1_1_1_1"><span class="lab"><font color="Green" title="E69">A98</font></span></a>, <a class="txt" href="#E4:15_4_1_1_1_1"><span class="lab"><font color="Green" title="E70">A99</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_4_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_4_1_1_1_1"><span class="lab"><font color="Green" title="E71">A100</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="15_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_4_1_1_1_2"/>
<font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:15_4_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A101</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">and </span><br/><a NAME="E4:15_4_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A102</font></span>: 
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/><a NAME="E5:15_4_1_1_1_2"/>
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E18:15_4_1"><span class="lab"><font color="Green" title="E65">A94</font></span></a>, <a class="txt" href="#E19:15_4_1"><span class="lab"><font color="Green" title="E66">A95</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:15_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E12:15_4_1"><span class="lab"><font color="Green" title="E62">A91</font></span></a>, <a class="txt" href="#E3:15_4_1_1_1_2"><span class="lab"><font color="Green" title="E69">A101</font></span></a>, <a class="txt" href="#E4:15_4_1_1_1_2"><span class="lab"><font color="Green" title="E70">A102</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/><a NAME="E7:15_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A103</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> <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="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E8:15_4_1_1_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E10:15_4_1"><span class="lab"><font color="Green" title="E60">A89</font></span></a>, <a class="txt" href="#E16:15_4_1"><span class="lab"><font color="Green" title="E64">A93</font></span></a>, <a class="txt" href="#E3:15_4_1_1_1_2"><span class="lab"><font color="Green" title="E69">A101</font></span></a>, <a class="txt" href="#E4:15_4_1_1_1_2"><span class="lab"><font color="Green" title="E70">A102</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_4_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4_1_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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="E21:15_4_1"/>
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_4_1"><span class="lab"><font color="Green" title="E63">A92</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E22:15_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E20:15_4_1"><span class="lab"><font color="Green" title="E67">A96</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E23:15_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A104</font></span>: 
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E24:15_4_1"/>
<font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_4_1"><span class="lab"><font color="Green" title="E59">A88</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E25:15_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E20:15_4_1"><span class="lab"><font color="Green" title="E67">A96</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E26:15_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E23:15_4_1"><span class="lab"><font color="Green" title="E68">A104</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:15_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_4_1"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="txt" href="#E15:15_4_1"><span class="lab"><font color="Green" title="E63">A92</font></span></a>, <a class="txt" href="#E9:15_4_1"><span class="lab"><font color="Green" title="E59">A88</font></span></a>, <a class="ref" href="seq_4.html#T155" title="SEQ_4:th.155">SEQ_4:155</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="E2:15_4"/><span class="lab"><font color="Green" title="E54">A105</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</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="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</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="E3:15_4"/><span class="lab"><font color="Green" title="E55">A106</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> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:15"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A107</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <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="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_4"><span class="lab"><font color="Green" title="E54">A105</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/>
<a NAME="E5:15_4"/>
<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="c3">D1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="integra3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E6:15_4"/><span class="kw">then </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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_4"><span class="lab"><font color="Green" title="E55">A106</font></span></a>, <a class="txt" href="#E2:15_4"><span class="lab"><font color="Green" title="E54">A105</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/>
<a NAME="E7:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A108</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="#E4:15_4"><span class="lab"><font color="Green" title="E56">A107</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E8:15_4"/><span class="lab"><font color="Green" title="E58">A109</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></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="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:15_4"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E10:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A110</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A111</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E12:15_4"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15_4"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E13:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A112</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<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="E14:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A113</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:15_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E16:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A114</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</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="E17:15_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E18:15_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_4"><span class="lab"><font color="Green" title="E53">A82</font></span></a></span>;<br/>
<a NAME="E19:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A115</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/>
<a NAME="E20:15_4"/>
 <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 
<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="E21:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A116</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E22:15_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E23:15_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A117</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="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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_4_2_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="E1:15_4_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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15_4"><span class="lab"><font color="Green" title="E66">A117</font></span></a>, <a class="txt" href="#E19:15_4"><span class="lab"><font color="Green" title="E64">A115</font></span></a>, <a class="txt" href="integra3.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_4_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_4_2_1_1"/><span class="lab"><font color="Green" title="E67">A118</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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"> <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:15_4_2_1_1"/>
 <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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</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="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E3:15_4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A119</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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E4:15_4_2_1_1"/>
 <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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E5:15_4_2_1_1"/><span class="kw">then </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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E6:15_4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A120</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="#E3:15_4_2_1_1"><span class="lab"><font color="Green" title="E68">A119</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/><a NAME="E7:15_4_2_1_1"/><span class="lab"><font color="Green" title="E70">A121</font></span>: 
 <a href="integra1.html#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</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="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></span>;<br/><a NAME="E8:15_4_2_1_1"/><span class="lab"><font color="Green" title="E71">A122</font></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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <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="txt" href="#E31:15"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="integra3.html#E11"><span class="lab"><font color="Green" title="E11">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E9:15_4_2_1_1"/><span class="lab"><font color="Green" title="E72">A123</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E10:15_4_2_1_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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/><a NAME="E11:15_4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A124</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E1:15_4_2_1_1"><span class="lab"><font color="Green" title="E67">A118</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E12:15_4_2_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E13:15_4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A125</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="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_4_2_1_1"><span class="lab"><font color="Green" title="E72">A123</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E14:15_4_2_1_1"/>
 <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="c4">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E11:15_4_2_1_1"><span class="lab"><font color="Green" title="E73">A124</font></span></a>, <a class="txt" href="#E9:15_4_2_1_1"><span class="lab"><font color="Green" title="E72">A123</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E15:15_4_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A126</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_4_2_1_1"><span class="lab"><font color="Green" title="E73">A124</font></span></a>, <a class="txt" href="#E6:15_4_2_1_1"><span class="lab"><font color="Green" title="E69">A120</font></span></a>, <a class="txt" href="#E13:15_4_2_1_1"><span class="lab"><font color="Green" title="E74">A125</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/><a NAME="E16:15_4_2_1_1"/>
<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></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="c5">g</font>,<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/><a NAME="E17:15_4_2_1_1"/><span class="kw">then </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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E1:15_4_2_1_1"><span class="lab"><font color="Green" title="E67">A118</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">then </span> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
<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="c5">g</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="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:15_4"><span class="lab"><font color="Green" title="E65">A116</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <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="c3">D1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E11:15_4_2_1_1"><span class="lab"><font color="Green" title="E73">A124</font></span></a>, <a class="txt" href="#E15:15_4_2_1_1"><span class="lab"><font color="Green" title="E75">A126</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4"><span class="lab"><font color="Green" title="E57">A108</font></span></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E19:15_4_2_1_1"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4_2_1_1"><span class="lab"><font color="Green" title="E70">A121</font></span></a>, <a class="txt" href="#E8:15_4_2_1_1"><span class="lab"><font color="Green" title="E71">A122</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="15_4_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_4_2_1_2"/><span class="lab"><font color="Green" title="E67">A127</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span></span><br/><div class="add"><a NAME="E2:15_4_2_1_2"/><span class="lab"><font color="Green" title="E68">A128</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></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="c5">g</font>,<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/><a NAME="E3:15_4_2_1_2"/><span class="lab"><font color="Green" title="E69">A129</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</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">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="txt" href="#E1:15_4_2_1_2"><span class="lab"><font color="Green" title="E67">A127</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</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">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E3:15_4_2_1_2"><span class="lab"><font color="Green" title="E69">A129</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E5:15_4_2_1_2"/><span class="kw">then </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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_4_2_1_2"><span class="lab"><font color="Green" title="E67">A127</font></span></a>, <a class="txt" href="#E2:15_4_2_1_2"><span class="lab"><font color="Green" title="E68">A128</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/><a NAME="E6:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A130</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></span>;<br/><a NAME="E7:15_4_2_1_2"/><span class="lab"><font color="Green" title="E71">A131</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="E8:15_4_2_1_2"/>
 <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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</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="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E9:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A132</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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E10:15_4_2_1_2"/><span class="lab"><font color="Green" title="E73">A133</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <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="E11:15_4_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="txt" href="#E1:15_4_2_1_2"><span class="lab"><font color="Green" title="E67">A127</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/><a NAME="E12:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A134</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E13:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A135</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c4">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E14:15_4_2_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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E15:15_4_2_1_2"/><span class="kw">then </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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/><a NAME="E16:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A136</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_4_2_1_2"><span class="lab"><font color="Green" title="E72">A132</font></span></a>, <a class="txt" href="#E12:15_4_2_1_2"><span class="lab"><font color="Green" title="E74">A134</font></span></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></span>;<br/><a NAME="E17:15_4_2_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:15_4_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</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="E19:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A137</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</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="E20:15_4_2_1_2"/><span class="lab"><font color="Green" title="E78">A138</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A139</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="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E19:15_4_2_1_2"><span class="lab"><font color="Green" title="E77">A137</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E22:15_4_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E23:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A140</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="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_4_2_1_2"><span class="lab"><font color="Green" title="E77">A137</font></span></a>, <a class="txt" href="#E20:15_4_2_1_2"><span class="lab"><font color="Green" title="E78">A138</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E24:15_4_2_1_2"/><span class="lab"><font color="Green" title="E81">A141</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:15_4_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E26:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A142</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="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <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="#E13:15_4_2_1_2"><span class="lab"><font color="Green" title="E75">A135</font></span></a>, <a class="txt" href="#E24:15_4_2_1_2"><span class="lab"><font color="Green" title="E81">A141</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><a NAME="E27:15_4_2_1_2"/>
 <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="c4">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</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"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15_4_2_1_2"><span class="lab"><font color="Green" title="E75">A135</font></span></a>, <a class="txt" href="#E24:15_4_2_1_2"><span class="lab"><font color="Green" title="E81">A141</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></span>;<br/><span class="kw">then </span> <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:15_4_2_1_2"><span class="lab"><font color="Green" title="E82">A142</font></span></a>, <a class="txt" href="#E16:15_4_2_1_2"><span class="lab"><font color="Green" title="E76">A136</font></span></a>, <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="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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15_4_2_1_2"><span class="lab"><font color="Green" title="E80">A140</font></span></a>, <a class="txt" href="#E21:15_4_2_1_2"><span class="lab"><font color="Green" title="E79">A139</font></span></a></span>
;<br/><a NAME="E29:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A143</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></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/><a NAME="E30:15_4_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A144</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <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="c3">D1</font></span>)</span> <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="c5">g</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="c3">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/><a NAME="E31:15_4_2_1_2"/><span class="lab"><font color="Green" title="E85">A145</font></span>: 
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">ID2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">IR</font> = <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> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:15_4_2_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></span>;<br/>
<a NAME="E2:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A146</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:15"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="integra3.html#E12"><span class="lab"><font color="Green" title="E12">Lm4</font></span></a></span>;<br/>
<a NAME="E3:15_4_2_1_2_3"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4"><span class="lab"><font color="Green" title="E57">A108</font></span></a>, <a class="txt" href="#E30:15_4_2_1_2"><span class="lab"><font color="Green" title="E84">A144</font></span></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>;<br/>
<a NAME="E4:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15_4_2_1_2"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="txt" href="#E2:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E86">A146</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E5:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E6:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A147</font></span>: 
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c5">g</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="c3">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E7:15_4_2_1_2_3"/>
<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="c5">g</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="c3">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4_2_1_2"><span class="lab"><font color="Green" title="E71">A131</font></span></a>, <a class="txt" href="#E2:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E86">A146</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E8:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E87">A147</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A148</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E10:15_4_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E11:15_4"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E13:15_4"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="txt" href="#E8:15_4"><span class="lab"><font color="Green" title="E58">A109</font></span></a>, <a class="txt" href="#E14:15_4"><span class="lab"><font color="Green" title="E62">A113</font></span></a>, <a class="txt" href="#E1:15_4_2_1_2"><span class="lab"><font color="Green" title="E67">A127</font></span></a>, <a class="ref" href="finseq_6.html#T124" title="FINSEQ_6:th.124">FINSEQ_6:124</a></span>;<br/>
<a NAME="E11:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A149</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:15"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E12:15_4_2_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15_4"><span class="lab"><font color="Green" title="E63">A114</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="E13:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></span>;<br/>
<a NAME="E14:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:15"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="integra3.html#E12"><span class="lab"><font color="Green" title="E12">Lm4</font></span></a></span>;<br/>
<a NAME="E15:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A150</font></span>: 
<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="c5">g</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="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_4_2_1_2"><span class="lab"><font color="Green" title="E71">A131</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E16:15_4_2_1_2_3"/><span class="lab"><font color="Green" title="E91">A151</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E17:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <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="#E11:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E89">A149</font></span></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></span>;<br/>
<a NAME="E18:15_4_2_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A152</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:15"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="integra3.html#E12"><span class="lab"><font color="Green" title="E12">Lm4</font></span></a></span>;<br/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>) <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_4_2_1_2"><span class="lab"><font color="Green" title="E70">A130</font></span></a>, <a class="txt" href="#E16:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E91">A151</font></span></a>, <a class="txt" href="#E11:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E89">A149</font></span></a>, <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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:15_4"><span class="lab"><font color="Green" title="E65">A116</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>
;<br/>
<a NAME="E20:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15_4_2_1_2"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="txt" href="#E18:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E92">A152</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E21:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E22:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">g</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="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E23:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="c4">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E90">A150</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:15_4_2_1_2_3"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E25:15_4_2_1_2_3"/><span class="kw">then </span>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15_4_2_1_2_3"><span class="lab"><font color="Green" title="E88">A148</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:15_4_2_1_2_3"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E32:15_4_2_1_2"/>
<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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <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="txt" href="#E31:15"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="integra3.html#E11"><span class="lab"><font color="Green" title="E11">Lm3</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E33:15_4_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="integra3.html#E13"><span class="lab"><font color="Green" title="E13">Lm5</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><span class="kw">hence </span><a NAME="E34:15_4_2_1_2"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:15_4_2_1_2"><span class="lab"><font color="Green" title="E83">A143</font></span></a>, <a class="txt" href="#E31:15_4_2_1_2"><span class="lab"><font color="Green" title="E85">A145</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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>
<span class="kw">hence </span><a NAME="E25:15_4"/>
<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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</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="E48:15"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E49:15"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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> <font color="Maroon" title="c4">D2</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="E50:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A153</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</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="E51:15"/>
<font color="Maroon" title="c7">j1</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> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E52:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A154</font></span>: 
<font color="Maroon" title="c7">j1</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="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</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="E53:15"/><span class="lab"><font color="Green" title="E55">A155</font></span>: 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D1</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="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E54:15"/><span class="lab"><font color="Green" title="E56">A156</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_5"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_5"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 
<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="E3:15_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A157</font></span>: 
<font color="Maroon" title="c3">D1</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="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15"><span class="lab"><font color="Green" title="E37">A66</font></span></a>, <a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E53:15"><span class="lab"><font color="Green" title="E55">A155</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="E4:15_5"/><span class="lab"><font color="Green" title="E58">A158</font></span>: 
not <font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_5_1"/>
<font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_5_1"/>
<font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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/>
<span class="kw">hence </span><a NAME="E3:15_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_5"><span class="lab"><font color="Green" title="E57">A157</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</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="E5:15_5"/>
<font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15"><span class="lab"><font color="Green" title="E37">A66</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:15_5"/><span class="kw">then </span>
( <font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D1</font> or <font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:15_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_5"><span class="lab"><font color="Green" title="E58">A158</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:15_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">D2</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="c4">D2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E3:15_5"><span class="lab"><font color="Green" title="E57">A157</font></span></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 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>
<a NAME="E55:15"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E21">A6</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="E56:15"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></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="E57:15"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E57">A159</font></span>: <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:15"><span class="lab"><font color="Green" title="E56">A156</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
;<br/>
<a NAME="E59:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:15"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></span>;<br/>
<a NAME="E60:15"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A160</font></span>: <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:15"><span class="lab"><font color="Green" title="E38">A67</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
;<br/>
<a NAME="E62:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <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="c5">g</font>,<font color="Maroon" title="c3">D1</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="E63:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A161</font></span>: 
<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E64:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></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"><font color="Maroon" title="c3">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:15"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E21:15"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></span>;<br/>
<a NAME="E65:15"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E66:15"/><span class="kw">then </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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:15"><span class="lab"><font color="Green" title="E54">A154</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E67:15"/><span class="kw">then </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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:15"><span class="lab"><font color="Green" title="E53">A153</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E68:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A162</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:15"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E69:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font> <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="c5">g</font>,<font color="Maroon" title="c4">D2</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="E70:15"/><span class="kw">then </span>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A163</font></span>: <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:15"><span class="lab"><font color="Green" title="E60">A162</font></span></a>, <a class="txt" href="#E63:15"><span class="lab"><font color="Green" title="E59">A161</font></span></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>
;<br/>
<a NAME="E72:15"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:15"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E73:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A164</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>)
 
<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="E74:15"/><span class="lab"><font color="Green" title="E63">A165</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>)
 
<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="E75:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A166</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:15"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E76:15"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</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">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E74:15"><span class="lab"><font color="Green" title="E63">A165</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E77:15"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <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"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</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"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E44:15"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="txt" href="#E73:15"><span class="lab"><font color="Green" title="E62">A164</font></span></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:15"><span class="lab"><font color="Green" title="E48">A77</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
;<br/>
<a NAME="E79:15"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font>) <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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">lower_volume</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">D2</font>, <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:15"><span class="lab"><font color="Green" title="E64">A166</font></span></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E80:15"/>
<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="c5">g</font>,<font color="Maroon" title="c4">D2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c5">g</font>,<font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">g</font></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#K15" title="INTEGRA1:func.15">delta</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:15"><span class="lab"><font color="Green" title="E61">A163</font></span></a>, <a class="txt" href="#E47:15"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="txt" href="#E38:15"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="txt" href="#E58:15"><span class="lab"><font color="Green" title="E57">A159</font></span></a>, <a class="txt" href="#E61:15"><span class="lab"><font color="Green" title="E58">A160</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>
