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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, , <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E6:18"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/>
<b>set </b><font color="Maroon">FAB</font> = <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<b>set </b><font color="Maroon">FAC</font> = <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<b>set </b><font color="Maroon">FCB</font> = <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<a NAME="E8:18"/>
( <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>, <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:18"/>
<font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R3">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#D2">INTEGRA5:def 2</a></i>;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R1">is_upper_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R2">is_lower_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp;  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>;<br/>
<a NAME="E12:18"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, , <a class="ref" href="rfunct_1.html#T91">RFUNCT_1:91</a></i>;<br/>
<a NAME="E13:18"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i><a class="ref" href="integra5.html#T9">INTEGRA5:9</a></i>;<br/>
<b>consider </b><font color="Maroon">T1</font> being   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E15:18"/><i><font color="Green">E66</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="integra4.html#T11">INTEGRA4:11</a></i>;<br/>
<a NAME="E16:18"/>
 ex <font color="Olive">PS2</font> being  <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>(  <a href="integra2.html#K2">delta</a> <font color="Olive">PS2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">PS2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  ex <font color="Olive">K</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E2:18_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>then consider </b><font color="Maroon">T</font> being   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E67</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">Tn</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">Tn</font> <a href="integra4.html#R1">divide_into_equal</a> <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">Tn</font> ) ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E68</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><b>consider </b><font color="Maroon">Tn</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">Tn</font> <a href="integra4.html#R1">divide_into_equal</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Tn</font> )
 <b>by </b><i><a class="txt" href="integra6.html#E1:18"><i><font color="Green">E27</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">Tn</font> = <font color="Maroon">Tn</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> ;<br/><b>take </b><font color="Maroon">Tn</font> = <font color="Maroon">Tn</font>;<br/><b>thus </b><a NAME="E5:18_1_1"/>
<font color="Maroon">Tn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="integra6.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E6:18_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="integra6.html#T3" target="_self">Th3</a>, <a class="ref" href="integra4.html#D1">INTEGRA4:def 1</a></i>;<br/><a NAME="E7:18_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tn</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E8:18_1_1"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tn</font>
 <b>by </b><i><a class="ref" href="integra6.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:18_1"/>
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  ex <font color="Olive">K</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> ) )
 <b>by </b><i><a class="txt" href="integra6.html#E1:18"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">PS2</font> being   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E18:18"/><i><font color="Green">E70</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">PS2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">PS2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  ex <font color="Olive">K</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">K</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:18"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> )
 <b>by </b><i><a class="txt" href="integra6.html#E1:18"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">e</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span> );<br/>
<a NAME="E21:18"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">D</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">D</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> ;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>thus </b><a NAME="E2:18_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">D</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">S2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E23:18"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="integra6.html#T3" target="_self">Th3</a>);<br/></i>
<b>reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">S2</font> as   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> ;<br/>
<a NAME="E25:18"/><i><font color="Green">E74</font></i>: 
( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> ) ) &amp;  <a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:18_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:18_3_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">e</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_3_1"/>
 ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, <a class="ref" href="integra6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:18_3"/>
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:18_3_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:18_3_2_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">PS2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="integra6.html#E1:18"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_3_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:18_3_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:18_3_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:18_3_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">PS2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="integra6.html#E2:18"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E5:18_3_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p3">(<span class="default"><font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/><a NAME="E6:18_3_2_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">e1</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Olive">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">K</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:18_3_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_3_2"/>
 <a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:18_3_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="ref" href="integra6.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_3"/>
( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S2i</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S2i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">S2i</font> ) ) &amp;  <a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">S</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> <a href="rfinseq.html#K1">/^</a> 1 );<br/>
<a NAME="E26:18"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">T</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">T</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">S</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E2:18_4"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">T</font> = <font color="Maroon">S</font> <a href="rfinseq.html#K1">/^</a> 1;<br/>

<a NAME="E3:18_4"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E4:18_4"/><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:18_4"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E6:18_4"/><b>then </b><i><font color="Green">E85</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E7:18_4"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">T</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="integra1.html#T36">INTEGRA1:36</a></i>;<br/>
<a NAME="E8:18_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T36">FINSEQ_5:36</a></i>;<br/>
<a NAME="E9:18_4"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E2:18"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:18_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E11:18_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:18_4"/><b>then </b>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E13:18_4"/><b>then </b>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E14:18_4"/><b>then </b>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E15:18_4"/><b>then </b>
<font color="Maroon">T</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E16:18_4"/>
( <font color="Maroon">T</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">T</font> is    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">T</font>] )
 <b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#D4">INTEGRA1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">T2</font> being   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E28:18"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E29:18"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">D</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">E</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E2:18_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="rfinseq.html#K1">/^</a> 1 )
 <b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<a NAME="E4:18_5"/>
 ex <font color="Olive">E1</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">E1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">E1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:18_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font> &amp; 2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E6:18_5"/>
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:18"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">D</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">D</font> holds <br/><font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">D</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">E</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E2:18_6"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="rfinseq.html#K1">/^</a> 1 )
 <b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">E1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E4:18_6"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">E1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E1</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<a NAME="E6:18_6"/><i><font color="Green">E95</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E7:18_6"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:18_6"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E9:18_6"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:18_6"/><b>then </b><i><font color="Green">E97</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:18_6"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E12:18_6"/><i><font color="Green">E116</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E13:18_6"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E14:18_6"/><b>then </b><i><font color="Green">E117</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:18_6"/><b>then </b>
<font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:18_6"/><b>then </b><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> 1 &amp; <font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E17:18_6"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E18:18_6"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_6_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:18_6_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:18_6_1"/><b>then </b><i><font color="Green">E122</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_6_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:18_6_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:18_6_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_6_1_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div>
<b>end;</b></div><b>hence </b><a NAME="E5:18_6_1"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra6.html#T7" target="_self">Th7</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:18_6"/>
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">XCB</font> =  <a href="rfunct_1.html#K5">chi</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<a NAME="E31:18"/><i><font color="Green">E124</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7"><b>proof </b></a><div class="add">

<div><i><font color="Green">E125</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:18_7_1"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:18_7_1"/><b>then </b><i><font color="Green">E126</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:18_7_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:18_7_1_1"/><i><font color="Green">E128</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra3.html#T8">INTEGRA3:8</a></i>;<br/><b>assume </b><a NAME="E2:18_7_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:18_7_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">S2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:18_7_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/><a NAME="E5:18_7_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E6:18_7_1_1"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:18_7_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_7_1_1_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:18_7_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E7:18_7_1_1_1"/><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E8:18_7_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">E</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E10:18_7_1_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="rfinseq.html#K1">/^</a> 1 )
 <b>by </b><i><a class="ref" href="integra6.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">E1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E12:18_7_1_1_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">E1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E13:18_7_1_1_1"/><i><font color="Green">E138</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E1</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:18_7_1_1_1"/><b>then </b><i><font color="Green">E139</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<div><i><font color="Green">E140</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_7_1_1_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E142</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E3:18_7_1_1_1_1"/><i><font color="Green">E143</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E145</font></i>:  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/><a NAME="E6:18_7_1_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E7:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/><a NAME="E8:18_7_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E9:18_7_1_1_1_1"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E10:18_7_1_1_1_1"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:18_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:18_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E13:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:18_7_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E15:18_7_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E16:18_7_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a>, , </i>;<br/><a NAME="E17:18_7_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:18_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:18_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E20:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E21:18_7_1_1_1_1"/><i><font color="Green">E149</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E22:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:18_7_1_1_1_1"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E24:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, , </i>;<br/><a NAME="E25:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E26:18_7_1_1_1_1"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E27:18_7_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, , </i>;<br/><a NAME="E28:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, , </i>;<br/><a NAME="E29:18_7_1_1_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E30:18_7_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E31:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p0">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, </i>;<br/><a NAME="E32:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/><a NAME="E33:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/><a NAME="E34:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E35:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E36:18_7_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E37:18_7_1_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_7_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_7_1_1_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:18_7_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:18_7_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T13">FINSEQ_3:13</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E6:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E8:18_7_1_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E10:18_7_1_1_1_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E11:18_7_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , <a class="ref" href="finseq_1.html#T81">FINSEQ_1:81</a></i>;<br/><a NAME="E12:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:18_7_1_1_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, </i>;<br/><i><font color="Green">E163</font></i>:  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
;<br/><a NAME="E15:18_7_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E16:18_7_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E17:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T81">FINSEQ_1:81</a></i>;<br/><a NAME="E18:18_7_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E19:18_7_1_1_1_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E20:18_7_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="integra6.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E21:18_7_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:18_7_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E23:18_7_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E24:18_7_1_1_1_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:18_7_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E7:18_7_1_1"/>
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E8:18_7_1_1"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E9:18_7_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E10:18_7_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_7_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_7"/>
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:18_7"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">S1</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> ex <font color="Olive">S2</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">S2</font> );<br/>
<a NAME="E32:18"/><i><font color="Green">E168</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">T</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>,<font color="Olive">T</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i0</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<b>take </b><font color="Maroon">T</font> = <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font>;<br/>

<a NAME="E3:18_8"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">S1</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S2</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E4:18_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E5:18_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="treal_1.html#T2">TREAL_1:2</a></i>;<br/>
<a NAME="E6:18_8"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E7:18_8"/>
 ex <font color="Olive">D</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> )
 
<b>by </b><i><a class="txt" href="integra6.html#E2:18"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E8:18_8"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 
;<br/>
<a NAME="E9:18_8"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E173</font></i>: <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> = 
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
<b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
<b>by </b><i>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_8_2"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">T</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:18_8_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:18_8_2"/><b>then </b><i><font color="Green">E175</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_8_2_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><a NAME="E2:18_8_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:18_8_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:18_8_2_1"/><b>then </b><i><font color="Green">E178</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:18_8_2_1"/><b>then </b>
( <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:18_8_2_1"/>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_8_2_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><a NAME="E2:18_8_2_2"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/><a NAME="E3:18_8_2_2"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:18_8_2_2"/><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:18_8_2_2"/>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E8:18_8_2_2"/><b>then </b><i><font color="Green">E183</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:18_8_2_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E10:18_8_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E11:18_8_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E12:18_8_2_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E185</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_8_2_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><a NAME="E2:18_8_2_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:18_8_2_2_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:18_8_2_2_1"/><b>then </b>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:18_8_2_2_1"/><b>then </b>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, </i>;<br/><a NAME="E6:18_8_2_2_1"/><b>then </b><i><font color="Green">E187</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:18_8_2_2_1"/>
 ex <font color="Olive">DD</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> st <br/>( <font color="Olive">DD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">DD</font> holds <br/><font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">DD</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> ) )
 <b>by </b><i><a class="txt" href="integra6.html#E3:18"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E8:18_8_2_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">S2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E9:18_8_2_2_1"/>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_8_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_8_2_2_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><a NAME="E2:18_8_2_2_2"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/><a NAME="E3:18_8_2_2_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:18_8_2_2_2"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:18_8_2_2_2"/><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:18_8_2_2_2"/>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E9:18_8_2_2_2"/><b>then </b><i><font color="Green">E194</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:18_8_2_2_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E11:18_8_2_2_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:18_8_2_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E13:18_8_2_2_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E14:18_8_2_2_2"/>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , <a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:18_8_2_2"/>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:18_8_2"/>
<font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">T</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E12:18_8"/><b>then </b>
<font color="Maroon">T</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E13:18_8"/><b>then </b>
<font color="Maroon">T</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a>, <a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:18_8"/>
( <font color="Maroon">T</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">T</font> is    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">i0</font>,<font color="Maroon">T</font>] )
 <b>by </b><i><a class="ref" href="integra1.html#D4">INTEGRA1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">T</font> being   <a href="integra2.html#NM1">DivSequence</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E34:18"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>,<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="integra6.html#E5:18"><i><font color="Green">E37</font></i></a>);<br/></i>
<a NAME="E35:18"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S0</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>  st <font color="Olive">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S0</font></span>)</span> holds <br/> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S0</font> be    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>;<br/>



<b>assume </b><a NAME="E1:18_9"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S0</font></span>)</span> )
 ;<br/>

<a NAME="E2:18_9"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:18_9"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a>, <a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:18"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S0</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>  st <font color="Olive">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S0</font></span>)</span> holds <br/> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S0</font> be    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>



<b>assume </b><a NAME="E1:18_10"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S0</font></span>)</span> )
 ;<br/>

<a NAME="E2:18_10"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:18_10"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a>, <a class="ref" href="integra6.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E37:18"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S0</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>  st <font color="Olive">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S0</font></span>)</span> holds <br/> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S0</font> be    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>;<br/>



<b>assume </b><a NAME="E1:18_11"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>assume </b><a NAME="E2:18_11"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S0</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">S1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <font color="Maroon">S2</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E4:18_11"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E6:18_11"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:18_11"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, </i>;<br/>
<a NAME="E8:18_11"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E9:18_11"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E10:18_11"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E11:18_11"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:18_11"/><i><font color="Green">E207</font></i>: 
(  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<div><i><font color="Green">E208</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_11_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>thus </b> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_11_2"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:18_11_2"/><i><font color="Green">E211</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:18_11_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/><a NAME="E4:18_11_2"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:18_11_2"/><i><font color="Green">E213</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:18_11_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T81">NAT_1:81</a></i>;<br/><a NAME="E7:18_11_2"/><b>then </b><i><font color="Green">E214</font></i>: 
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:18_11_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E9:18_11_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:18_11_2"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:18_11"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:18"/><i><font color="Green">E216</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S01</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span><br/> for <font color="Olive">S02</font> being   <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>  st <font color="Olive">S01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">S02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S02</font></span>)</span> holds <br/> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">S01</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S01</font> be    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>;<br/><b>let </b><font color="Maroon">S02</font> be    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>





<b>assume </b><a NAME="E1:18_12"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">S01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">S02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:18_12"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S02</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">S1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <font color="Maroon">S2</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E4:18_12"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E6:18_12"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:18_12"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:18_12"/><i><font color="Green">E224</font></i>: 
(  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<div><i><font color="Green">E225</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_12_1"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:18_12_1"/><b>then </b><i><font color="Green">E227</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:18_12_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:18_12_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E5:18_12_1"/><b>then </b><i><font color="Green">E228</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_12_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:18_12_2"/><i><font color="Green">E230</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:18_12_2"/><b>then </b><i><font color="Green">E231</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:18_12_2"/><b>then </b><i><font color="Green">E232</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:18_12_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/><a NAME="E6:18_12_2"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:18_12_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T81">NAT_1:81</a></i>;<br/><a NAME="E8:18_12_2"/><b>then </b><i><font color="Green">E234</font></i>: 
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:18_12_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E10:18_12_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:18_12_2"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b> <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p4">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:18_12"/>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S01</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, <a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">XAC</font> =  <a href="rfunct_1.html#K5">chi</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<b>set </b><font color="Maroon">XAB</font> =  <a href="rfunct_1.html#K5">chi</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>;<br/>
<a NAME="E39:18"/><i><font color="Green">E238</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_13"><b>proof </b></a><div class="add">

<div><i><font color="Green">E239</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:18_13_1"/><i><font color="Green">E240</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/><b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:18_13_1"/><i><font color="Green">E241</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:18_13_1"/><i><font color="Green">E243</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, <a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="square_1.html#T49">SQUARE_1:49</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_13_1_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:18_13_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E3:18_13_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:18_13_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/><a NAME="E5:18_13_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/><a NAME="E6:18_13_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T8">INTEGRA3:8</a></i>;<br/><a NAME="E7:18_13_1_1"/><b>then </b><i><font color="Green">E246</font></i>: 
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E8:18_13_1_1"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/><a NAME="E9:18_13_1_1"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:18_13_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/><a NAME="E11:18_13_1_1"/><b>then </b><i><font color="Green">E247</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/><a NAME="E12:18_13_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T8">INTEGRA3:8</a></i>;<br/><a NAME="E13:18_13_1_1"/><b>then </b><i><font color="Green">E248</font></i>: 
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E14:18_13_1_1"/><i><font color="Green">E249</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:18_13_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_13_1_1_1"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:18_13_1_1_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Tm</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E7:18_13_1_1_1"/><i><font color="Green">E253</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tm</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E8:18_13_1_1_1"/><b>then </b><i><font color="Green">E254</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">S1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <font color="Maroon">S2</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E10:18_13_1_1_1"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E11:18_13_1_1_1"/><i><font color="Green">E256</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tm</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E12:18_13_1_1_1"/><i><font color="Green">E257</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Tm</font> )
 
<b>by </b><i>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_13_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_13_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_13_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_13_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><div class="add"><a NAME="E2:18_13_1_1_1_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:18_13_1_1_1_1_1"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E4:18_13_1_1_1_1_1"/><i><font color="Green">E260</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E5:18_13_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:18_13_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:18_13_1_1_1_1_1"/><b>then </b><i><font color="Green">E261</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E8:18_13_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E9:18_13_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:18_13_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, </i>;<br/><a NAME="E11:18_13_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E12:18_13_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_13_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_13_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 ;<br/><div class="add"><a NAME="E2:18_13_1_1_1_1_2"/><b>then </b><i><font color="Green">E262</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:18_13_1_1_1_1_2"/><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:18_13_1_1_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:18_13_1_1_1_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>set </b><font color="Maroon">i1</font> = 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>;<br/><a NAME="E8:18_13_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="integra6.html#T16" target="_self">Th16</a>, <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:18_13_1_1_1_1_2"/><b>then </b><i><font color="Green">E265</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:18_13_1_1_1_1_2"/><b>then </b><i><font color="Green">E266</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E11:18_13_1_1_1_1_2"/><i><font color="Green">E267</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E12:18_13_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:18_13_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p5">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="integra6.html#E15:18"><i><font color="Green">E66</font></i></a>, , </i>;<br/><a NAME="E14:18_13_1_1_1_1_2"/><b>then </b><i><font color="Green">E268</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E15:18_13_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E16:18_13_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E17:18_13_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, </i>;<br/><a NAME="E18:18_13_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>hence </b><a NAME="E19:18_13_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:18_13_1_1"/>
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>,<span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E16:18_13_1_1"/><b>then </b><i><font color="Green">E269</font></i>: 
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/><a NAME="E17:18_13_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T8">INTEGRA3:8</a></i>;<br/><a NAME="E18:18_13_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E19:18_13_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:18_13_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_13"/>
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:18_13"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:18"/><i><font color="Green">E270</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14"><b>proof </b></a><div class="add">

<a NAME="E1:18_14"/><i><font color="Green">E271</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">F</font> =  <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F1</font> =  <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F2</font> =  <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">S1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <font color="Maroon">S2</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E5:18_14_1"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<i><font color="Green">E275</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>
;<br/>
<a NAME="E8:18_14_1"/><b>then </b><i><font color="Green">E276</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green">E277</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:18_14_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:18_14_1_2"/><b>then </b><i><font color="Green">E278</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><b>assume </b><a NAME="E3:18_14_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/><a NAME="E4:18_14_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:18_14_1_2"/><b>then </b><i><font color="Green">E279</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E6:18_14_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E7:18_14_1_2"/><b>then </b><i><font color="Green">E280</font></i>: 
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E8:18_14_1_2"/><i><font color="Green">E281</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T7" target="_self">Th7</a>, , </i>;<br/><a NAME="E9:18_14_1_2"/><i><font color="Green">E282</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, , , <a class="ref" href="integra6.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E10:18_14_1_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:18_14_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E12:18_14_1_2"/>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_14_1_3"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font>
 ;<br/><a NAME="E2:18_14_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:18_14_1_3"/><b>then </b><i><font color="Green">E284</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:18_14_1_3"/><b>then </b><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:18_14_1_3"/><i><font color="Green">E286</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:18_14_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:18_14_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:18_14_1_3"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:18_14_1_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:18_14_1_3"/><b>then </b><i><font color="Green">E288</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E11:18_14_1_3"/><i><font color="Green">E289</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E12:18_14_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E13:18_14_1_3"/><b>then </b><i><font color="Green">E290</font></i>: 
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E14:18_14_1_3"/><i><font color="Green">E291</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T8" target="_self">Th8</a>, , , <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E15:18_14_1_3"/><i><font color="Green">E292</font></i>: 
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, , <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E16:18_14_1_3"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:18_14_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a>, </i>;<br/><b>hence </b><a NAME="E18:18_14_1_3"/>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:18_14_1"/>
 <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_14"/><i><font color="Green">E293</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:18_14_2"/>
 <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E2:18_14_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_14_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:18_14"/><b>then </b><i><font color="Green">E294</font></i>: 
 <a href="integra2.html#K3">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<a NAME="E5:18_14"/><i><font color="Green">E295</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> &amp;  <a href="integra2.html#K3">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="integra4.html#T9">INTEGRA4:9</a></i>;<br/>
<a NAME="E6:18_14"/>
<font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#T9">INTEGRA5:9</a></i>;<br/>
<a NAME="E7:18_14"/><b>then </b>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra4.html#T9">INTEGRA4:9</a></i>;<br/>
<b>hence </b><a NAME="E8:18_14"/>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E296</font></i>:  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> = 
 <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
<b>by </b><i>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E42:18"/><i><font color="Green">E297</font></i>: 
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>;<br/>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> = 
 <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
<b>by </b><i>, <a class="ref" href="integra5.html#D5">INTEGRA5:def 5</a></i>
<br/>.= 
 <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>

;<br/>
<b>hence </b><a NAME="E44:18"/>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T10" target="_self">Th10</a>, </i>;<br/>

<a NAME="E45:18"/><i><font color="Green">E298</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17"><b>proof </b></a><div class="add">

<a NAME="E1:18_17"/><i><font color="Green">E299</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">F</font> =  <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F1</font> =  <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F2</font> =  <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">S1</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>, <font color="Maroon">S2</font> being    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span><b> such that </b><br/><a NAME="E5:18_17_1"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="integra6.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="integra1.html#M1">DivisionPoint</a> of <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <b>by </b><i><a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<i><font color="Green">E301</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E8:18_17_1"/><b>then </b><i><font color="Green">E302</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:18_17_1"/><b>then </b><i><font color="Green">E303</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green">E304</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_17_1_2"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/><a NAME="E2:18_17_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:18_17_1_2"/><b>then </b><i><font color="Green">E306</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E4:18_17_1_2"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:18_17_1_2"/><b>then </b><i><font color="Green">E308</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E6:18_17_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E7:18_17_1_2"/><b>then </b><i><font color="Green">E309</font></i>: 
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E8:18_17_1_2"/><i><font color="Green">E310</font></i>: 
(  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> &amp;  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> )
 <b>by </b><i><a class="ref" href="integra6.html#T6" target="_self">Th6</a>, <a class="ref" href="integra6.html#T7" target="_self">Th7</a>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E9:18_17_1_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:18_17_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E11:18_17_1_2"/>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:18_17_1_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font>
 ;<br/><a NAME="E2:18_17_1_3"/><b>then </b><i><font color="Green">E311</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:18_17_1_3"/><b>then </b><i><font color="Green">E312</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:18_17_1_3"/><b>then </b><i><font color="Green">E313</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:18_17_1_3"/><i><font color="Green">E314</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:18_17_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:18_17_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E9:18"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:18_17_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E7:18"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E9:18_17_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E10:18_17_1_3"/><b>then </b><i><font color="Green">E315</font></i>: 
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E11:18_17_1_3"/><i><font color="Green">E316</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S1</font> )
 <b>by </b><i>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E12:18_17_1_3"/><b>then </b><i><font color="Green">E317</font></i>: 
(  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> &amp;  <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 <b>by </b><i><a class="txt" href="integra6.html#E7"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="integra6.html#T8" target="_self">Th8</a>, <a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E13:18_17_1_3"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:18_17_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p5"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p5">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E11:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra6.html#T16" target="_self">Th16</a>, </i>;<br/><b>hence </b><a NAME="E15:18_17_1_3"/>
<font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a>, <a class="txt" href="integra6.html#E13:18"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:18_17_1"/>
 <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18_17"/><i><font color="Green">E318</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:18_17_2"/>
 <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E2:18_17_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_17_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra6.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p4"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:18_17"/><b>then </b><i><font color="Green">E319</font></i>: 
 <a href="integra2.html#K4">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<a NAME="E5:18_17"/><i><font color="Green">E320</font></i>: 
(  <a href="integra2.html#K4">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> &amp;  <a href="integra2.html#K4">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra6.html#E4:18"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="integra6.html#T1" target="_self">Th1</a>, <a class="ref" href="integra4.html#T8">INTEGRA4:8</a></i>;<br/>
<a NAME="E6:18_17"/>
<font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 
<b>by </b><i>, <a class="ref" href="integra5.html#T9">INTEGRA5:9</a></i>;<br/>
<a NAME="E7:18_17"/><b>then </b>
 <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra6.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra4.html#T8">INTEGRA4:8</a></i>;<br/>
<b>hence </b><a NAME="E8:18_17"/>
 <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra6.html#T15" target="_self">Th15</a>, , <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:18"/>
(  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> &amp;  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T51">INTEGRA1:51</a></i>;<br/>
<a NAME="E47:18"/><b>then </b><i><font color="Green">E321</font></i>: 
(  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span></span>)</span> &amp;  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E48:18"/>
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R1">is_upper_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R2">is_lower_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R1">is_upper_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R2">is_lower_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i>, , <a class="ref" href="integra4.html#T10">INTEGRA4:10</a></i>;<br/>
<a NAME="E49:18"/><b>then </b>
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R3">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R3">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 
<b>by </b><i><a class="ref" href="integra6.html#T11" target="_self">Th11</a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E50:18"/>
( <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> )
 <b>by </b><i><a class="ref" href="integra5.html#D2">INTEGRA5:def 2</a></i>;<br/>


</div>
