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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c1">A</font><br/>  for <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font>  st  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b3">i</font>) &amp; ( <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Olive" title="b1">D</font>,<font color="Olive" title="b3">i</font>)</span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font>  st  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b2">i</font>) &amp; ( <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b2">i</font>)</span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E17">AS</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E18">A24</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E19">A25</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra3.html#T3" onmouseover="rs('integra3/T3')" onmouseout="rh()">INTEGRA3:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_1_1"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E19">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E20">A26</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )</span><br/><div class="add"><a NAME="E2:16_1_2"/>
<font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI: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#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E19">A25</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/><a NAME="E3:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A30</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="kw">thus </span><a NAME="E4:16_1_2"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_2_1_1"/>
(  not  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> or  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_1_1"/>
 not  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )</span><br/><div class="add"><a NAME="E2:16_1_2_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E21">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E19">A25</font></span></a>, <a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E21">A30</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_1_2"/><span class="lab"><font color="Green" title="E22">A31</font></span>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )</span><br/><div class="add"><a NAME="E2:16_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">A38</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">j</font> = <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><a NAME="E4:16_1_2_1_1_2"/><span class="lab"><font color="Green" title="E24">A40</font></span>: 
( <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_2_1_1_2_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
;<br/>
<a NAME="E2:16_1_2_1_1_2_1"/><span class="kw">then </span>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:16_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E25">A41</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A31</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span><br/>

<a NAME="E4:16_1_2_1_1_2_1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T11" onmouseover="rs('seq_4/T11')" onmouseout="rh()">SEQ_4:11</a></span>;<br/>
<a NAME="E5:16_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI: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#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E25">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:16_1_2_1_1_2"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">j</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="16_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E25">A42</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:16_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_2_1_1_2_2_1_1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E17">AS</font></span></a>, <a class="txt" href="#E1:16_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E25">A42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E25">A44</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:16_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A45</font></span>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E3:16_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="txt" href="#E1:16_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E25">A44</font></span></a>, <a class="ref" href="integra1.html#T7" onmouseover="rs('integra1/T7')" onmouseout="rh()">INTEGRA1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_2_1_1_2_2_1_2"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E26">A45</font></span></a>, <a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E6:16_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="txt" href="#E4:16_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">A40</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
