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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:51"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:51"/><b>then </b>
 <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font>, B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> : ( ex <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></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">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> ) or ( not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> ) ) ) ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="chain_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:51"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> iff ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; ( ex <font color="Olive">b<sub>4</sub></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">c<sub>2</sub></font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></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">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> ) or ( not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>5</sub></font> ) ) ) ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>4</sub></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">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>4</sub></font> ) ) ) ) ) )
 ;<br/>


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