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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> 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>, <font color="Olive" title="b3">D1</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b2">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> holds <br/> <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D1</font>,<font color="Olive" title="b2">D</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D1</font>,<font color="Olive" title="b2">D</font>,<font color="Maroon" title="c2">j</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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>, <font color="Olive" title="b2">D1</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c3">A</font>  st <font color="Olive" title="b1">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b2">D1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> holds <br/> <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D1</font>,<font color="Olive" title="b1">D</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D1</font>,<font color="Olive" title="b1">D</font>,<font color="Maroon" title="c2">j</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">D1</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">j</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font>
 <span class="kw">and </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">and </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">j</font></span><br/>

<a NAME="E5:9"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E6:9"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E7:9"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E8:9"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E9:9"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:9"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</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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