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<b>let </b><font color="Maroon" title="c5">A1</font>, <font color="Maroon" title="c6">A2</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> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:20_1_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E2:20_1_2_1"/><i><font color="Green" title="E14">A12</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> )
 <b>and </b><br/><a NAME="E3:20_1_2_1"/><i><font color="Green" title="E15">A13</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:20_1_2_1"/><i><font color="Green" title="E16">A14</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/><a NAME="E6:20_1_2_1"/>
<font color="Maroon" title="c5">A1</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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:20_1_2_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="integra1.html#E5:20_1_2_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E7:20_1_2_1"/>
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:20_1_2_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="integra1.html#E5:20_1_2_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:20_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E3:20_1_2"/><i><font color="Green" title="E14">A15</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> )
 <b>and </b><br/><a NAME="E4:20_1_2"/><i><font color="Green" title="E15">A16</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:20_1_2"/><i><font color="Green" title="E16">A17</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E7:20_1_2"/><i><font color="Green" title="E17">A18</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/>
<a NAME="E8:20_1_2"/>
<font color="Maroon" title="c5">A1</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"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:20_1_2"><i><font color="Green" title="E14">A15</font></i></a>, <a class="txt" href="integra1.html#E6:20_1_2"><i><font color="Green" title="E16">A17</font></i></a>, <a class="txt" href="integra1.html#E7:20_1_2"><i><font color="Green" title="E17">A18</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E9:20_1_2"/>
<font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:20_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="integra1.html#E6:20_1_2"><i><font color="Green" title="E16">A17</font></i></a>, <a class="txt" href="integra1.html#E7:20_1_2"><i><font color="Green" title="E17">A18</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>


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