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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">D1</font>, <font color="Olive" title="b5">D2</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">D1</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">D1</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="Olive" title="b4">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b5">D2</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="b5">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b5">D2</font>,<font color="Olive" title="b4">D1</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b5">D2</font>,<font color="Olive" title="b4">D1</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b6">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>  ex <font color="Olive" title="b7">MD1</font>, <font color="Olive" title="b8">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b6">B</font> 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="b6">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">MD2</font> <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> <font color="Olive" title="b8">MD2</font></span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">MD1</font> <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> <font color="Olive" title="b7">MD1</font></span>)</span> &amp; <font color="Olive" title="b7">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b8">MD2</font> &amp; <font color="Olive" title="b7">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b4">D1</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b8">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b5">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b5">D2</font>,<font color="Olive" title="b4">D1</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b5">D2</font>,<font color="Olive" title="b4">D1</font>,<font color="Olive" title="b2">j</font></span>)</span> )</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">D1</font>, <font color="Olive" title="b3">D2</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">D1</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">D1</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="Olive" title="b2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">D2</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="b3">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D2</font>,<font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D2</font>,<font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 holds <br/> ex <font color="Olive" title="b4">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>  ex <font color="Olive" title="b5">MD1</font>, <font color="Olive" title="b6">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b4">B</font> 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="b4">B</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">MD2</font> <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> <font color="Olive" title="b6">MD2</font></span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">MD1</font> <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> <font color="Olive" title="b5">MD1</font></span>)</span> &amp; <font color="Olive" title="b5">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b6">MD2</font> &amp; <font color="Olive" title="b5">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b6">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b3">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D2</font>,<font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b3">D2</font>,<font color="Olive" title="b2">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> )</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">D1</font>, <font color="Olive" title="b2">D2</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">D1</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">D1</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="Olive" title="b1">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b2">D2</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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b4">MD1</font>, <font color="Olive" title="b5">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b3">B</font> 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> <font color="Olive" title="b5">MD2</font> <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> <font color="Olive" title="b5">MD2</font></span>)</span> &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> <font color="Olive" title="b4">MD1</font> <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> <font color="Olive" title="b4">MD1</font></span>)</span> &amp; <font color="Olive" title="b4">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b5">MD2</font> &amp; <font color="Olive" title="b4">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b5">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Olive" title="b2">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">D1</font>, <font color="Maroon" title="c6">D2</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">D1</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">D1</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="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">MD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">MD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E27">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">D1</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">D1</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="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> ) )</span><br/>

<a NAME="E2:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:26"/><span class="lab"><font color="Green" title="E29">A3</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">D1</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="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:26"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> ) )</span><br/>

<a NAME="E5:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:26"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<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="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</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 = 
<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"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</font>

;<br/>
<span class="kw">assume </span><a NAME="E9:26"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c5">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> ) )</span><br/>

<a NAME="E10:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c6">D2</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="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span> <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> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E11:26"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c6">D2</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="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <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> <font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E12:26"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E14:26"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>;<br/>
<a NAME="E15:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<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="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E16:26"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/>
<a NAME="E17:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A15</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E36">A10</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">assume </span><a NAME="E18:26"/><span class="lab"><font color="Green" title="E42">A16</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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E20:26"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<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="Maroon" title="c9">B</font>
 <span class="kw">and </span><br/><a NAME="E21:26"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c9">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="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></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="c6">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:26"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="integra3.html#T11" target="_self" title="INTEGRA3:th.11">Th11</a></span>;<br/>
<a NAME="E23:26"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="integra1.html#T60" title="INTEGRA1:th.60">INTEGRA1:60</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">MD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c9">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E45">A20</font></span></a></span>;<br/>
<a NAME="E25:26"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E26:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E48">A22</font></span>: <font color="Maroon" title="c10">MD2</font> <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> <font color="Maroon" title="c10">MD2</font></span>)</span> = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>
;<br/>
<font color="Maroon" title="c10">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="E30:26"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<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="Maroon" title="c11">C</font>
 <span class="kw">and </span><br/><a NAME="E31:26"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c11">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">D1</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">D1</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="E32:26"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</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="c11">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="integra3.html#T11" target="_self" title="INTEGRA3:th.11">Th11</a></span>;<br/>
<a NAME="E33:26"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="integra1.html#T60" title="INTEGRA1:th.60">INTEGRA1:60</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A27</font></span>: <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</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">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span> = 
<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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">j</font>

;<br/>
<span class="lab"><font color="Green" title="E54">A28</font></span>: <font color="Maroon" title="c9">B</font> = 
<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="c9">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="c9">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/>.= 
<font color="Maroon" title="c11">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E30:26"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E31:26"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="txt" href="#E27:26"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">MD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c9">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E51">A26</font></span></a></span>;<br/>
<a NAME="E37:26"/><span class="lab"><font color="Green" title="E55">A29</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> or <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">MD2</font> )</span><br/>

<a NAME="E1:26_6"/><span class="lab"><font color="Green" title="E56">A30</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T125" title="FINSEQ_6:th.125">FINSEQ_6:125</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_6"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</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="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">MD2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k1</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> <span class="kw"> such that </span><br/><a NAME="E4:26_6"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
<font color="Maroon" title="c14">k1</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="c12">MD1</font>
 <span class="kw">and </span><br/><a NAME="E5:26_6"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<a NAME="E6:26_6"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<a NAME="E7:26_6"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_6"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">k2</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> <span class="kw"> such that </span><br/><a NAME="E9:26_6"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c15">k2</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="c6">D2</font>
 <span class="kw">and </span><br/><a NAME="E10:26_6"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_6"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<a NAME="E11:26_6"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<font color="Maroon" title="c14">k1</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="c12">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E12:26_6"/><span class="lab"><font color="Green" title="E63">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E13:26_6"/><span class="kw">then </span>
1 <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="c12">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_6"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:26_6"/><span class="kw">then </span>
1 <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="c12">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E15:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E12:26_6"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/>
<a NAME="E16:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A38</font></span>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E9:26_6"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E10:26_6"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E5:26_6"><span class="lab"><font color="Green" title="E59">A33</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">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k3</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:26_6"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c15">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">k3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E19:26_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</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="c12">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E20:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">MD1</font> <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> <font color="Maroon" title="c12">MD1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E11:26_6"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="seq_4.html#T154" title="SEQ_4:th.154">SEQ_4:154</a></span>;<br/>
<a NAME="E21:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c15">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E34:26"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="txt" href="#E9:26_6"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E10:26_6"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E5:26_6"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E22:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c15">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E23:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E24:26_6"/>
<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_6"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E25:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A41</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<a NAME="E26:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</font></span>: 
<font color="Maroon" title="c16">k3</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="c10">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E23:26_6"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E18:26_6"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E27:26_6"/>
<font color="Maroon" title="c16">k3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E28:26_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E25:26_6"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="txt" href="#E23:26_6"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E18:26_6"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="finseq_6.html#T128" title="FINSEQ_6:th.128">FINSEQ_6:128</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:26_6"/>
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_6"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E18:26_6"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E26:26_6"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E38:26"/>
<font color="Maroon" title="c10">MD2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
;<br/>
<a NAME="E39:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c10">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></span>;<br/>
<a NAME="E40:26"/>
<font color="Maroon" title="c12">MD1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
;<br/>
<a NAME="E41:26"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></span>;<br/>
<a NAME="E42:26"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</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="c10">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:26"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E43:26"/><span class="kw">then </span>
<font color="Maroon" title="c12">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c10">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:26"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E44:26"/>
 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>  ex <font color="Olive" title="b2">MD1</font>, <font color="Olive" title="b3">MD2</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Olive" title="b1">B</font> 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> <font color="Olive" title="b3">MD2</font> <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> <font color="Olive" title="b3">MD2</font></span>)</span> &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> <font color="Olive" title="b2">MD1</font> <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> <font color="Olive" title="b2">MD1</font></span>)</span> &amp; <font color="Olive" title="b2">MD1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b3">MD2</font> &amp; <font color="Olive" title="b2">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> &amp; <font color="Olive" title="b3">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c3">j</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E31:26"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E35:26"><span class="lab"><font color="Green" title="E54">A28</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/>


</div>
