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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="b3">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Olive" title="b3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <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> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b3">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b3">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c3">A</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <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> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">B</font> holds <br/> <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <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> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">B</font> implies  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">B</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">and </span><br/><a NAME="E4:65"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <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> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E5:65"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E6:65"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">i</font></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> <font color="Maroon" title="c2">j</font>
 
;<br/>
<a NAME="E7:65"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E45">A1</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="E8:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</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="E9:65"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E10:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E11:65"/>
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E46">A2</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="E12:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<font color="Maroon" title="c2">j</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="c5">D</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/>
<span class="kw">consider </span><font color="Maroon" title="c6">A1</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> <span class="kw"> such that </span><br/><a NAME="E14:65"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E15:65"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</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> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:65"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="integra1.html#T38" target="_self" title="INTEGRA1:th.38">Th38</a></span>;<br/>
<a NAME="E17:65"/><span class="lab"><font color="Green" title="E57">A12</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> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="integra1.html#E59"><span class="lab"><font color="Green" title="E42">Th38a</font></span></a></span>;<br/>
<a NAME="E18:65"/><span class="lab"><font color="Green" title="E58">A14</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">i</font></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> <font color="Maroon" title="c1">i</font>
 
;<br/>
<a NAME="E19:65"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E2:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c6">A1</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></span>;<br/>
<a NAME="E3:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">A1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <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="c6">A1</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E4:65_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A15</font></span>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">A1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <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="c6">A1</font> )
 
;<br/>
<a NAME="E5:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E14:65"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E18:65"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>;<br/>
<a NAME="E6:65_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A16</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:65_1"/>
<font color="Maroon" title="c7">x</font> <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="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E17:65"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>;<br/>
<a NAME="E8:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <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="c4">B</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a></span>;<br/>
<a NAME="E9:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">B</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:65_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</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="E20:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/>
<a NAME="E21:65"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:65"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></span>;<br/>
<a NAME="E22:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A18</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:65"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:65"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</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> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E17:65"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:65"/>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E16:65"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E22:65"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</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>
