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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<a NAME="E1:20"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:20_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:20_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E3:20_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">RFUNCT_1:73</a></i>;<br/>
<a NAME="E5:20_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E3:20_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E6:20_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:20_1_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E4:20_1_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra2.html#T8" target="_self" title="INTEGRA2:th.8">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20"/>
<font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>

<a NAME="E2:20_2_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="integra2.html#T8" target="_self" title="INTEGRA2:th.8">Th8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:20_2_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:20_2_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E4:20_2_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E7:20_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E6:20_2_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E8:20_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:20_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:20_2_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="integra2.html#E6:20_2_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">RFUNCT_1:73</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_2"/>
<font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
