<?xml version="1.0"?>
<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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:71"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<i><font color="Green">E39</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E4:71"/>
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"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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">X</font>;<br/>

<b>assume </b><a NAME="E1:71_3"/><i><font color="Green">E45</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="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:71_3"/><b>then </b><i><font color="Green">E48</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E3:71_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="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E4:71_3"/><b>then </b><i><font color="Green">E49</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="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_2.html#T46">FUZZY_2:46</a></i>
;<br/>
<b>hence </b><a NAME="E7:71_3"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:71"/>
 <a href="rfunct_3.html#K19">max+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">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/>

<a NAME="E6:71"/>
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"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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">X</font>;<br/>

<b>assume </b><a NAME="E1:71_4"/><i><font color="Green">E77</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="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:71_4"/><b>then </b><i><font color="Green">E78</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<a NAME="E3:71_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="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E4:71_4"/><b>then </b><i><font color="Green">E79</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="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="rfunct_3.html#K3">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K3">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_2.html#T46">FUZZY_2:46</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 0</span>)</span>

;<br/>
<b>hence </b><a NAME="E7:71_4"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:71"/>
 <a href="rfunct_3.html#K20">max-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">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>
