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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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">G</font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">d</font>;<br/>





<b>assume </b><a NAME="E1:50"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E2:50"/><b>then </b>
 <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">l</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</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>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">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="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</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">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) ) ) ) or ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</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> holds <br/> ( <font color="Olive">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="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</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">i</font> ) ) ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font></span>)</span> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> iff  ex <font color="Olive">l</font>, <font color="Olive">r</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">l</font>,<font color="Olive">r</font> &amp; (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</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>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">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="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</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">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) ) ) ) or ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</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> holds <br/> ( <font color="Olive">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="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</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">i</font> ) ) ) ) ) )
 ;<br/>


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