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<b>let </b><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:102"/><i><font color="Green" title="E74">A1</font></i>: 
<font color="Maroon" title="c1">F1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 ;<br/>

<b>assume </b><a NAME="E2:102"/><i><font color="Green" title="E75">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">F2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<b>assume </b><a NAME="E3:102"/>
<font color="Maroon" title="c2">F2</font> is <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>
 ;<br/>

<a NAME="E4:102"/><b>then </b><i><font color="Green" title="E76">A3</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c2">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#D24" target="_self" title="SUPINF_2:def.24">Def24</a></i>;<br/>
<a NAME="E5:102"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c2">F2</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:102"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="supinf_2.html#E2:102"><i><font color="Green" title="E75">A2</font></i></a>, <a class="ref" href="supinf_2.html#T62" target="_self" title="SUPINF_2:th.62">Th62</a></i>;<br/>
<a NAME="E6:102"/><b>then </b><i><font color="Green" title="E77">A4</font></i>: 
not  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_2.html#E4:102"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<a NAME="E7:102"/><i><font color="Green" title="E78">A5</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:102"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="supinf_2.html#T59" target="_self" title="SUPINF_2:th.59">Th59</a></i>;<br/>
<a NAME="E8:102"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:102"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a></i>;<br/>
<a NAME="E10:102"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E7:102"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:102"/><b>then </b><i><font color="Green" title="E79">A6</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>;<br/>
<a NAME="E12:102"/>
(  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:102"/>
<font color="Maroon" title="c1">F1</font> is <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>
 <b>by </b><i><a class="txt" href="supinf_2.html#E6:102"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="supinf_2.html#E11:102"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="supinf_2.html#D24" target="_self" title="SUPINF_2:def.24">Def24</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


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