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<div class="add">

<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="E1:98"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:98"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E3:98"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:98_1"/><b>then </b><i><font color="Green">E75</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> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98"><i><font color="Green">E73</font></i></a>, <a class="ref" href="supinf_2.html#T63" target="_self">Th63</a></i>;<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> 0 as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:98_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <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>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="supinf_1.html#T33">SUPINF_1:33</a></i>;<br/>
<a NAME="E6:98_1"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="supinf_2.html#E4:98_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:98"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:98_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:98_2"/><i><font color="Green">E76</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>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:98_2_1"/>
 <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#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>3</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:98"><i><font color="Green">E72</font></i></a>, <a class="ref" href="supinf_2.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E2:98_2_1"/>
<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>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</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>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:98_2"/><i><font color="Green">E77</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>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>3</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>5</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E6:98_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:98"><i><font color="Green">E73</font></i></a>, <a class="txt" href="supinf_2.html#E1:98_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="supinf_2.html#E2:98_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="supinf_2.html#E4:98_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="supinf_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:98_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17" target="_self">rng</a> <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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E8:98_2"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17" target="_self">rng</a> <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>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E6:98_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="supinf_1.html#T33">SUPINF_1:33</a></i>;<br/>
<a NAME="E9:98_2"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" target="_self">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E6:98_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="supinf_2.html#E7:98_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="supinf_1.html#T80">SUPINF_1:80</a></i>;<br/>
<b>hence </b><a NAME="E10:98_2"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:98"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>


</div>
