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<div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">s1</font> =  <a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font>;<br/>
<a NAME="E2:69"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> ;<br/>
<a NAME="E3:69"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="real_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E3:69_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E4:69_1"/>
<font color="Maroon">S<sub>2</sub></font>[1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:69"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:69_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E4:69_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n3</font> = <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:69_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E7:69_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2]
 <b>by </b><i><a class="txt" href="real_3.html#E1:69_2"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="fib_num.html#S1">FIB_NUM:sch 1</a>(<a class="ref" href="real_3.html#T11" target="_self">Th11</a>, , <a class="ref" href="real_3.html#D2" target="_self">Def2</a>);<br/></i>
<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E7:69"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E8:69"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK9" target="_self">c_n</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div>
