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<b>let </b><font color="Maroon">S1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">S2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:79_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="supinf_2.html#K13" target="_self">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:79_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="supinf_2.html#K13" target="_self">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:79_1_1"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <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/>
<a NAME="E4:79_1_1_1"/>
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">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:79_1_1_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_1_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:79_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font>, <font color="Maroon">y2</font> = <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">k</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>thus </b><font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">y2</font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="supinf_2.html#E21"><i><font color="Green">Lemma62</font></i></a></i>
<br/>.= 
<font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:79_1_1_1_1"/>
for <font color="Olive">k</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">k</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#E20"><i><font color="Green">Lemma61</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E5:79_1_1_1"/><b>then </b>
<font color="Maroon">S1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E6:79_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:79_1_1"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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