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<b>let </b><font color="Maroon">F1</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">F2</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="E1:101"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F1</font> is <a href="supinf_2.html#V6" target="_self">nonnegative</a>
 ;<br/>

<b>assume </b><a NAME="E2:101"/><i><font color="Green">E28</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">F1</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F2</font> <a href="supinf_2.html#K13" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<b>assume </b><a NAME="E3:101"/>
<font color="Maroon">F2</font> is <a href="supinf_2.html#V7" target="_self">summable</a>
 ;<br/>

<a NAME="E4:101"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F2</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#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E5:101"/>
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:101"/><b>then </b><i><font color="Green">E30</font></i>: 
not  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<a NAME="E7:101"/><i><font color="Green">E31</font></i>: 
 <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">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E8:101"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <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">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" target="_self">Ser</a> <font color="Maroon">F1</font></span>)</span> <a href="supinf_2.html#K13" target="_self">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <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">F1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<a NAME="E10:101"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E11:101"/><i><font color="Green">E51</font></i>: 
 <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</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/>
<a NAME="E12:101"/>
(  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  <a href="supinf_2.html#K19" target="_self">SUM</a> <font color="Maroon">F1</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="E13:101"/>
<font color="Maroon">F1</font> is <a href="supinf_2.html#V7" target="_self">summable</a>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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