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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:106"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:106"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E3:106"/><i><font color="Green">E38</font></i>: 
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>
 <b>and </b><br/><a NAME="E4:106"/><i><font color="Green">E39</font></i>: 
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>
 <b>and </b><br/><a NAME="E5:106"/><i><font color="Green">E40</font></i>: 
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>
 <b>and </b><br/><a NAME="E6:106"/><i><font color="Green">E41</font></i>: 
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>
 <b>and </b><br/><a NAME="E7:106"/><i><font color="Green">E42</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">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> )
 ;<br/>

<b>reconsider </b><font color="Maroon">DG</font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="txt" href="mesfunc5.html#E18:106"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">E0</font> =  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">E'</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span>;<br/>
<a NAME="E9:106"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> holds <br/><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>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D16">MESFUNC1:def 16</a></i>;<br/>
<a NAME="E10:106"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">DG</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E44</font></i>: <font color="Maroon">DG</font> = 
<font color="Maroon">DG</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/>
<a NAME="E12:106"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">DG</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/>
<a NAME="E13:106"/><b>then </b>
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18">great_eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T31">MESFUNC1:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">GG</font> = <font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K18">great_eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E15:106"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">GG</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/>
<a NAME="E16:106"/><b>then </b>
<font color="Maroon">GG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16">less_eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T32">MESFUNC1:32</a></i>;<br/>
<a NAME="E17:106"/><b>then </b>
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T22">MESFUNC1:22</a></i>;<br/>
<a NAME="E18:106"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<a href="supinf_2.html#K1">0.</a>  <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E19:106"/><b>then </b>
 <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E20:106"/><b>then </b>
 <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/>
<a NAME="E21:106"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a></i>;<br/>
<a NAME="E22:106"/>
<font color="Maroon">DG</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/>
<a NAME="E23:106"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a></i>;<br/>
<font color="Maroon">DG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">DG</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<font color="Maroon">DG</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">E'</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E0</font> =  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E27:106"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">E0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">E'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><a NAME="E28:106"/>
 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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><b>proof </b></a><div class="add">

<a NAME="E1:106_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">E'</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<i><font color="Green">E100</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E'</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">E'</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E3:106_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:106_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="mesfunc5.html#E11:106"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E4:106_3"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:106_3_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 ;<br/>

<a NAME="E2:106_3_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DG</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:106_3_2"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D16">MESFUNC1:def 16</a></i>;<br/>
<a NAME="E4:106_3_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:106_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<b>hence </b><a NAME="E5:106_3_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font>;<br/>
<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="E6:106_3"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<div><i><font color="Green">E106</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:106_3_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:106_3_3"/>
<font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:106_3"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> (  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> &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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_4"/>
 <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E'</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E0</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E2:106_3_4"><i><font color="Green">E108</font></i></a></i>;<br/>
<a NAME="E2:106_3_4"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">E'</font> <a href="tarski.html#R1">c=</a>  <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="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>thus </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E'</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>

<b>hence </b><a NAME="E4:106_3_4"/>
 <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:106_3"/><i><font color="Green">E109</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</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"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> &amp; <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_5"><b>proof </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="E1:106_3_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>thus </b><a NAME="E2:106_3_5"/>
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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</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"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> &amp; <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="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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_5_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="E1:106_3_5_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 ;<br/>

<a NAME="E2:106_3_5_1"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E3:106_3_5_1"/>
<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> <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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:106_3_5"><i><font color="Green">E121</font></i></a>, , <a class="txt" href="mesfunc5.html#E1:106_3_5_1"><i><font color="Green">E122</font></i></a>, , <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E4:106_3_5_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:106_3_5_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:106_3_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:106_3_5_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>

<a NAME="E6:106_3_5_1"/><b>then </b>
<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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:106_3_5_1"/>
<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> <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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E10:106_3"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> 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="partfun1.html#K2">|</a> <font color="Maroon">E'</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_6"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:106_3"><i><font color="Green">E102</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:106_3_6"/>
<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="partfun1.html#K2">|</a> <font color="Maroon">E'</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:106"><i><font color="Green">E44</font></i></a></i>;<br/>

<b>hence </b><a NAME="E3:106_3_6"/>
<font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:106_3"/><i><font color="Green">E125</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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> 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; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <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="106_3_7"><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="E1:106_3_7"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 ;<br/>

<a NAME="E2:106_3_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:106_3_7"/><b>then </b><i><font color="Green">E127</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">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:106_3_7"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">F'</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:106_3_7_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/><b>thus </b><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> = 
<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>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:106_3_7"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:106_3_7_1"/>
<font color="Maroon">F'</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:106_3_7"/>
( <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="txt" href="mesfunc5.html#E10:106_3"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E7:106_3"><i><font color="Green">E106</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:106_3_7"/>
( <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; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <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="txt" href="mesfunc5.html#E11:106_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:106_3"/><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:106_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   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">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
;<br/>
<a NAME="E14:106_3"/><b>then </b>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E15:106_3"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:106_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="mesfunc5.html#E3:106_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, </i>;<br/>
<a NAME="E16:106_3"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E0</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:106_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="mesfunc5.html#E3:106_3"><i><font color="Green">E101</font></i></a>, </i>;<br/>
<a NAME="E17:106_3"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</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">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:106_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</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="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E19:106_3"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:106_3_8"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:106_3_8"/>
<font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E21:106_3"/><i><font color="Green">E135</font></i>: 
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/><font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_9"><b>proof </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="E1:106_3_9"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:106_3_9"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:106_3"><i><font color="Green">E102</font></i></a></i>;<br/>
<a NAME="E3:106_3_9"/><i><font color="Green">E138</font></i>: 
(  <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:106_3_9"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:106_3"><i><font color="Green">E104</font></i></a></i>;<br/>
<a NAME="E5:106_3_9"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><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="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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:106_3_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:106_3_9_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:106_3_9_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:106_3_9_1"/>
<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> <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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:106_3"><i><font color="Green">E131</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:106_3_9"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E12:106_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E17:106_3"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E7:106_3_9"/><b>then </b><i><font color="Green">E141</font></i>: 
( <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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> &amp; <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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E15:106_3"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E8:106_3_9"/>
( <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">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="Maroon">n</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</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="Maroon">m</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:106_3_9"/>
<font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:106_3"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E12:106_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E17:106_3"><i><font color="Green">E132</font></i></a>, <a class="txt" href="mesfunc5.html#E21:106"><i><font color="Green">E94</font></i></a>, <a class="txt" href="mesfunc5.html#E20:106_3"><i><font color="Green">E134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E22:106_3"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">G</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="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E23:106_3"/><i><font color="Green">E143</font></i>: 
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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_10"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:106_3"><i><font color="Green">E104</font></i></a></i>;<br/>
<a NAME="E2:106_3_10"/><b>then </b>
<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="partfun1.html#K2">|</a> <font color="Maroon">E'</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:106_3_10"/>
<font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E25:106_3"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E185</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:106_3_11"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:106_3_11"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:106_3"><i><font color="Green">E131</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:106_3"/><i><font color="Green">E186</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> 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>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E28:106_3"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">L</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:106_3"><i><font color="Green">E132</font></i></a></i>
<br/>.= 
 <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="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E29:106_3"/><b>then </b><i><font color="Green">E188</font></i>: 
( <font color="Maroon">L</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>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:106"><i><font color="Green">E98</font></i></a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, , <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="txt" href="mesfunc5.html#E12:106_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E19:106_3"><i><font color="Green">E133</font></i></a>, </i>;<br/>
<a NAME="E30:106_3"/>
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/><font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_13"><b>proof </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="E1:106_3_13"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:106_3_13"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:106_3_13"/><i><font color="Green">E191</font></i>: 
(  <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> &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="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E4:106_3_13"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:106_3"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E5:106_3_13"/><i><font color="Green">E193</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><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="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="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:106_3_13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:106_3_13_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:106_3_13_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:106_3_13_1"/>
<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> <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="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#T31" target="_self">Th31</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:106_3_13"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:106_3_13"/><b>then </b><i><font color="Green">E194</font></i>: 
( <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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> &amp; <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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<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#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, </i>;<br/>
<a NAME="E8:106_3_13"/>
( <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">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="Maroon">n</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</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="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:106_3"><i><font color="Green">E134</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:106_3_13"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , <a class="txt" href="mesfunc5.html#E21:106"><i><font color="Green">E94</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:106_3"/><b>then </b><i><font color="Green">E195</font></i>: 
( <font color="Maroon">L</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">L</font> <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> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E32:106_3"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:106_3_14"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">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="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:106_3_14"/><i><font color="Green">E197</font></i>: 
 <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E'</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E0</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:106_3_14"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:106_3"><i><font color="Green">E102</font></i></a></i>;<br/>
<a NAME="E4:106_3_14"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:106_3"><i><font color="Green">E104</font></i></a></i>;<br/>
<a NAME="E5:106_3_14"/><b>then </b><i><font color="Green">E199</font></i>: 
( <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="partfun1.html#K2">|</a> <font color="Maroon">E0</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <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="partfun1.html#K2">|</a> <font color="Maroon">E'</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:106_3_14"/>
( <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="partfun1.html#K2">|</a> <font color="Maroon">E0</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <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="partfun1.html#K2">|</a> <font color="Maroon">E'</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E11:106"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E7:106_3_14"/><b>then </b><i><font color="Green">E200</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E0</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, </i>;<br/>
<a NAME="E8:106_3_14"/><i><font color="Green">E201</font></i>: 
 <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:106_3_14"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <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="Maroon">n</font></span>)</span> holds <br/><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="Olive">x</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="Olive">x</font>
 
;<br/>
<a NAME="E10:106_3_14"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:106_3"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E11:106_3_14"/><b>then </b><i><font color="Green">E203</font></i>: 
<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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">E'</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:106_3"><i><font color="Green">E104</font></i></a></i>;<br/>
<a NAME="E12:106_3_14"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:106_3"><i><font color="Green">E102</font></i></a></i>;<br/>
<a NAME="E13:106_3_14"/><b>then </b>
 <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E0</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E23:106_3"><i><font color="Green">E143</font></i></a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, </i>;<br/>
<a NAME="E14:106_3_14"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T20">SUPINF_2:20</a></i>;<br/>
<b>hence </b><a NAME="E15:106_3_14"/>
<font color="Maroon">L</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E17:106_3"><i><font color="Green">E132</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:106_3"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E34:106_3"/>
( ( 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="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">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="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E21:106_3"><i><font color="Green">E135</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a>, <a class="txt" href="mesfunc5.html#E11:106_3"><i><font color="Green">E125</font></i></a>, , , </i>;<br/>


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


</div>
