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<b>let </b><font color="Maroon" title="c1">p</font> be  <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>   st <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="Maroon" title="c1">p</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> holds <br/> <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> is  <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><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c1">p</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> implies  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is  <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><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E44">A1</font></i>: 
<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="c1">p</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E45">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is  <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><br/>

<a NAME="E3:63"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:63"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="jordan3.html#D1" title="JORDAN3:def.1">JORDAN3:def 1</a></i>;<br/>
<a NAME="E4:63"/>
<font color="Maroon" title="c3">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="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:63"/><b>then </b>
<font color="Maroon" title="c3">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="c1">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:63"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E2:63"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:63"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> is  <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> 
 
<b>by </b><i><a class="ref" href="integra1.html#T36" target="_self" title="INTEGRA1:th.36">Th36</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is  <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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a>, <a class="ref" href="goboard2.html#T20" title="GOBOARD2:th.20">GOBOARD2:20</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> is  <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> 
 <b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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