<?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">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/>





<a NAME="E1:69"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E2:69"/><b>then </b><i><font color="Green">E37</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">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</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> <font color="Maroon">f</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> &amp;  <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">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</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> <font color="Maroon">f</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><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a>, <a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<a NAME="E3:69"/><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> <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> &amp;  <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> &amp;  <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">g</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"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">g</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><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a>, <a class="ref" href="rfunct_3.html#D11">RFUNCT_3:def 11</a></i>;<br/>
<a NAME="E4:69"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></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"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></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"><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#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E5:69"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E6:69"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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> )
 ;<br/>

 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<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#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E8:69"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<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#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">g</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> <font color="Maroon">f</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>
 ;<br/>

<a NAME="E9:69"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<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#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E10:69"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<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#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">g</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> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E11:69"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p3">(<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#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">g</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> <font color="Maroon">f</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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </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="E1:69_2"/><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"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:69_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <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">x</font> )
 <b>by </b><i><a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a>, <a class="ref" href="rfunct_3.html#T43">RFUNCT_3:43</a></i>;<br/><a NAME="E3:69_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:69_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:69"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <font color="Maroon">f</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>now </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="E1:69_3"/><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"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:69_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<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> )
 <b>by </b><i><a class="ref" href="rfunct_3.html#T40">RFUNCT_3:40</a>, <a class="ref" href="rfunct_3.html#T43">RFUNCT_3:43</a></i>;<br/><a NAME="E3:69_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:69_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</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>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:69"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K20">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/>


</div>
