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

<b>let </b><font color="Maroon">C</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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 ;<br/>

<i><font color="Green">E37</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<i><font color="Green">E38</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<a NAME="E4:39"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</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"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></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">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><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">C</font>;<br/>

<b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E39</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"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:39_3"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/>
<a NAME="E3:39_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E4:39_3"/><b>then </b><i><font color="Green">E41</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"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,0
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:39_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,0
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:39_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:39_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="mesfunc5.html#E3:39"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K1">max</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,<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="txt" href="mesfunc5.html#E5:39_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
 <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <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="txt" href="mesfunc5.html#E2:39"><i><font color="Green">E37</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E7:39_3"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39"/>
 <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:39"><i><font color="Green">E38</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>

<a NAME="E6:39"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</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"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></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">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><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">C</font>;<br/>

<b>assume </b><a NAME="E1:39_4"/><i><font color="Green">E43</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"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:39_4"/><b>then </b><i><font color="Green">E44</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">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E3:39_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E4:39_4"/><b>then </b><i><font color="Green">E47</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"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K1">max</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<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="mesfunc2.html#D3">MESFUNC2:def 3</a></i>
<br/>.= 
 <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <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="txt" href="mesfunc5.html#E2:39"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
 <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <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="extreal1.html#T26">EXTREAL1:26</a></i>
;<br/>
<b>hence </b><a NAME="E7:39_4"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:39_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:39"/>
 <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
