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<span class="kw">let </span><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#K7" title="NUMBERS:func.7">ExtREAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  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#K12" title="SUPINF_2:func.12">.</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#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) &amp; <font color="Maroon" title="c2">F2</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  implies <font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:69"/>
<font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <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>  st  not <font color="Maroon" title="c1">F1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> or  not <font color="Maroon" title="c2">F2</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  or <font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  )</span><br/>

<a NAME="E2:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 <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#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T40" target="_self" title="SUPINF_2:th.40">Th59</a></span>;<br/>
<a NAME="E3:69"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <font color="Maroon" title="c1">F1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/>
<a NAME="E4:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:69"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
 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#K12" title="SUPINF_2:func.12">.</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#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">F2</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  or <font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E6:69"/>
<font color="Maroon" title="c2">F2</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a> </span><br/>

<a NAME="E7:69"/><span class="kw">then </span>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c2">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D18" target="_self" title="SUPINF_2:def.18">Def24</a></span>;<br/>
<a NAME="E8:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 not  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="supinf_2.html#T43" target="_self" title="SUPINF_2:th.43">Th62</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<a NAME="E9:69"/>
(  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:69"/>
<font color="Maroon" title="c1">F1</font> is  <a href="supinf_2.html#V7" title="SUPINF_2:attr.7">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="supinf_2.html#D18" target="_self" title="SUPINF_2:def.18">Def24</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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