<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E34</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ] <b>means </b>( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E2:29"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">H</font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a> ;<br/>
<b>assume </b><a NAME="E1:29_1"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a> </span>)</span>
 ;<br/>

<b>thus </b> <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a> </span>)</span></span>)</span> = 
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:29"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">H</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="binarith.html#K11">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>]
 ;<br/>

<b>thus </b><a NAME="E2:29_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:29_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E3:29_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:29_2_1"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E5:29_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:29_2_1"/><i><font color="Green">E70</font></i>: 
(  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:29_2_1"/>
<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>hence </b> <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="convfun1.html#D1" target="_self">Def1</a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:29"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">H</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>assume </b><a NAME="E5:29"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 ;<br/>

<b>hence </b><a NAME="E6:29"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="convfun1.html#D1" target="_self">Def1</a></i>;<br/>


</div>
