<?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#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>thus </b><a NAME="E1:63"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> implies <font color="Maroon">f1</font> <a href="rfunct_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="63_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="rfunct_1.html#K4" target="_self">^</a>  is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:63_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_1.html#K4" target="_self">^</a> </span>)</span> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2" target="_self">/</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D4" target="_self">Def4</a></i>;<br/>


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

<b>assume </b><a NAME="E2:63"/>
<font color="Maroon">f1</font> <a href="rfunct_1.html#K2" target="_self">/</a> <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="rfunct_1.html#K4" target="_self">^</a> </span>)</span> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>;<br/>

<a NAME="E5:63"/>
<font color="Maroon">f2</font> <a href="rfunct_1.html#K4" target="_self">^</a>  is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E6:63"/>
( <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f2</font> is <a href="partfun1.html#V1">total</a> )
 <b>by </b><i><a class="ref" href="rfunct_1.html#T13" target="_self">Th13</a></i>;<br/>


</div>
