<?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#K1">REAL</a> ;<br/>



<a NAME="E1:27"/><i><font color="Green">E38</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<a NAME="E2:27"/>
 <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"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</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="E3:27"/><b>then </b><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"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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#D7">SEQ_1:def 7</a></i>;<br/>
<a NAME="E4:27"/>
for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then </b><i><font color="Green">E41</font></i>: <span class="p1">(<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">x1</font> = 
 <a href="rfunct_3.html#K3">max-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D2">RFUNCT_3:def 2</a></i>
;<br/>
<a NAME="E3:27_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> = 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</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"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,0
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E5:27_1"/>
<span class="p1">(<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">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:27"/>
 <a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K19">max+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</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>
