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<b>let </b><font color="Maroon">c<sub>1</sub></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><a NAME="E1:103"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</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> ] means <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> ;<br/>
<a NAME="E2:103"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</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>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E3:103"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:103"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green">E77</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="103_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:103_1"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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#T63" target="_self">Th63</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_2.html#E1:103"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_2.html#E1:103_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E6:103_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E7:103_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:103_1"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:103"/>
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#E3:103"><i><font color="Green">E76</font></i></a>, <a class="txt" href="supinf_2.html#E4:103"><i><font color="Green">E77</font></i></a>);</i><br/>


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