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<b>let </b><font color="Maroon">d</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">l</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">d</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1"/>
 <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> 1,<font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">l'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E37</font></i>: 
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font> &amp; ( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) ) &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 <b>by </b><i/>;<br/><a NAME="E4:61_1"/>
( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_1"/>
( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) )
 </span><b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:61_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E2:61_1_1_1_1_1"/>
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:61_1_1_1_1"/>
( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:61_1_1_1_2_1"/>
( 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> <font color="Maroon">d</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E2:61_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_2_1"/>
<font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:61_1_1_1_2"/>
( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:61_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E6:61_1"/>
 ex <font color="Olive">i0</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> st <br/>( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> ) ) &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i0</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i0</font> holds <br/>( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:61"/>
(  ex <font color="Olive">i0</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> st <br/>( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font> ) ) &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i0</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i0</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i0</font> holds <br/>( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) implies  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> 1,<font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div>
