<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds <br/> ( ( <font color="Olive" title="b1">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Olive" title="b2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> ) iff <font color="Olive" title="b1">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Olive" title="b2">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> ) iff <font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/>





<b>thus </b><a NAME="E1:60"/>
( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies <font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies ( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a></span><br/>

<a NAME="E2:60_1"/><b>then </b>
<font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a>  is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T74" target="_self" title="CFUNCT_1:th.74">Th74</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b>
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a> </span>)</span> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="#E1:60_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="cfunct_1.html#T70" target="_self" title="CFUNCT_1:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T51" target="_self" title="CFUNCT_1:th.51">Th51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:60"/>
<font color="Maroon" title="c2">f1</font> <a href="cfunct_1.html#K1" title="CFUNCT_1:func.1">/</a> <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/>

<a NAME="E3:60"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a> </span>)</span> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T51" target="_self" title="CFUNCT_1:th.51">Th51</a></i>;<br/>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon" title="c2">f1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T70" target="_self" title="CFUNCT_1:th.70">Th70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/>

<a NAME="E5:60"/>
<font color="Maroon" title="c3">f2</font> <a href="cfunct_1.html#K2" title="CFUNCT_1:func.2">^</a>  is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="cfunct_1.html#T70" target="_self" title="CFUNCT_1:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E6:60"/>
( <font color="Maroon" title="c3">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 <b>by </b><i><a class="ref" href="cfunct_1.html#T74" target="_self" title="CFUNCT_1:th.74">Th74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
