<?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">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<i><font color="Green">E24</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K10">"</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="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</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="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K10">"</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="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K10">"</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/>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:19"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f1</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><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="E1:19_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<a NAME="E2:19_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:19_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f1</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon">f1</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="mesfunc1.html#K5" target="_self">+</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
