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<span class="kw">let </span><font color="Maroon" title="c1">r</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">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">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="b4">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b3">A</font>  st <font color="Olive" title="b1">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="b4">D</font> &amp; <font color="Olive" title="b2">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="b4">D</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b4">D</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b5">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b5">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b5">B</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="Olive" title="b4">D</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="Olive" title="b4">D</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b4">D</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b5">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">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="c2">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="b2">D</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">D</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b3">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b3">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b3">B</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="Olive" title="b2">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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="Olive" title="b2">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b2">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b3">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">A</font>  st <font color="Maroon" title="c2">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="c3">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="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 holds <br/> ex <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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b2">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b2">B</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="Olive" title="b1">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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="Olive" title="b1">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b1">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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="c5">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c4">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="c2">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="c3">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="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 implies  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">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>
 ; <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">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> or  not <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> or  not <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 or  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:18"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c3">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>
 ; <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="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> or  not <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 or  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:18"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 or  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">C</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="E5:18"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">C</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="c2">i</font>,<font color="Maroon" title="c3">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="E6:18"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c6">C</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:18"/><span class="lab"><font color="Green" title="E23">A6</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="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="integra1.html#T38" title="INTEGRA1:th.38">INTEGRA1:38</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">MD</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:18"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> )</span><br/>

<a NAME="E10:18"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <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="c6">C</font> &amp; <font color="Olive" title="b2">b</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="c6">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></span>;<br/>
<a NAME="E11:18"/><span class="kw">then </span>
<font color="Maroon" title="c1">r</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">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<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">C</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <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">by</span> <span class="lab"><a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/>
<a NAME="E13:18"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c8">B</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> <font color="Maroon" title="c8">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="c8">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="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/>
<a NAME="E14:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<a NAME="E15:18"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c8">B</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="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<a NAME="E16:18"/>
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">C</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="c8">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E2:18_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></span>;<br/>
<a NAME="E3:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:18_1"/>
<font color="Maroon" title="c9">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="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E28">A11</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="E5:18_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</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="E17:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">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="E18:18"/>
 <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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<a NAME="E19:18"/><span class="kw">then </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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:18"/><span class="kw">then </span>
<font color="Maroon" title="c7">MD</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:18"/>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c1">r</font> <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="Olive" title="b1">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">B</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="c2">i</font>,<font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E27">A10</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/>


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