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

<b>consider </b><font color="Maroon">F</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:108_1_1"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) ) )
 <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> 0 as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<a NAME="E4:108_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:108_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:108_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:108_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:108_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E3:108_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E4:108_1_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:108_1_1_1"/>
( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:108_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:108_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds   <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> ) ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:108_1_1"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="mesfunc5.html#NM1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Olive">G</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Olive">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Olive">G</font> &amp;  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Olive">G</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E9:108_1_1"/><i><font color="Green">E44</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">G</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">G</font> )
 ;<br/>
<b>take </b>
 <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E10:108_1_1"/>
 ex <font color="Olive">F</font> being   <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  ex <font color="Olive">K</font> being  <a href="mesfunc5.html#NM1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( <font color="Olive">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Olive">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Olive">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Olive">K</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Olive">K</font> )
 <b>by </b><i>, </i>;<br/>


</div>
