<?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">V</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>







<b>thus </b><a NAME="E1:36"/>
( ( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> ) iff <font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_1"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> implies <font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> = 
<font color="Maroon">C</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">C</font>

;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:36_1"/>
<font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E3:36_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:36_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:36_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:36_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:36"/>
( ( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> ) iff <font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_2"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> implies <font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_2_1"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 ;<br/>

<a NAME="E2:36_2_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font></span>)</span> = 
<font color="Maroon">C</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">C</font>

;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:36_2"/>
<font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E3:36_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:36_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:36_2"/><b>then </b>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:36_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a><br/>


</div><b>end;</b></div>


</div>
