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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="E1:104"/><i><font color="Green">E75</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="E2:104"/><i><font color="Green">E76</font></i>: 
for <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="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><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#R1">=</a>  <a href="supinf_2.html#K1" target="_self">0.</a> 
 <b>and </b><br/><a NAME="E3:104"/><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="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E4:104"/><i><font color="Green">E78</font></i>: 
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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>2</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:104"><i><font color="Green">E75</font></i></a>, <a class="txt" href="supinf_2.html#E2:104"><i><font color="Green">E76</font></i></a>, <a class="ref" href="supinf_2.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E5:104"/>
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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:104_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:104_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</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#E3:104"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E3:104_1_1"/><i><font color="Green">E81</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>1</sub></font> <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#E1:104"><i><font color="Green">E75</font></i></a>, <a class="ref" href="supinf_2.html#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E4:104_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <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="txt" href="supinf_2.html#E3:104_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<a NAME="E5:104_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="supinf_1.html#K3">REAL</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a></i>;<br/>
<a NAME="E6:104_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:104_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:104_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="supinf_2.html#E4:104_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:104_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:104_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1" target="_self">0.</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:104"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:104_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:104_1"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:104"/><b>then </b>
<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> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T68" target="_self">Th68</a></i>;<br/>
<b>hence </b><a NAME="E7:104"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="supinf_2.html#V7" target="_self">summable</a>
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:104"><i><font color="Green">E78</font></i></a>, <a class="ref" href="supinf_2.html#D24" target="_self">Def24</a></i>;<br/>


</div>
