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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="supinf_2.html#NM2">Pos_Denum_Set_of_R_EAL</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="supinf_2.html#M1" target="_self">Num</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:81"/><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self">Def17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:81"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0</span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E2:81_1"/><i><font color="Green">E63</font></i>: 
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:81"><i><font color="Green">E60</font></i></a>, <a class="ref" href="supinf_2.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E3:81_1"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> 1
 
<b>by </b><i><a class="ref" href="supinf_2.html#T54" target="_self">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:81_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:81_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="supinf_2.html#E2:81_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:81"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:81_2"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E3:81_2"/><i><font color="Green">E65</font></i>: 
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E4:81_2"/>
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:81_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<b>hence </b><a NAME="E5:81_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:81_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="supinf_2.html#E3:81_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="supinf_2.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:81"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="supinf_2.html#E2:81"><i><font color="Green">E61</font></i></a>, <a class="txt" href="supinf_2.html#E3:81"><i><font color="Green">E62</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:81"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


</div>
