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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="lattice5.html#M2" target="_self">ExtensionSeq</a> of <font color="Maroon">A</font>,<font color="Maroon">d</font>;<br/><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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  );<br/>
<a NAME="E1:74"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E2:74"/><i><font color="Green">E37</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">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_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>assume </b><b>that </b><br/><a NAME="E1:74_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  )
 <b>and </b><br/><a NAME="E2:74_1"/><i><font color="Green">E42</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> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A'</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:74_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Aq</font>,<font color="Maroon">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E4:74_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">d'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E9:74_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of <font color="Maroon">d'</font><b> such that </b><br/><a NAME="E6:74_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Aq</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E7:74_1_1_2"/>
<font color="Maroon">dq</font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E6:74_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E8:74_1_1_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E9:74_1_1_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A'</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d'</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E10:74_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> <font color="Maroon">A'</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K8" target="_self">DistEsti</a> <font color="Maroon">d'</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:74_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:74"/>
for <font color="Olive">l</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">l</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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