<?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">f</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">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> iff  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Olive" title="b1">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/><b>let </b><font color="Maroon" title="c2">f</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">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> iff  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><br/>



<b>thus </b><a NAME="E1:57"/>
( <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</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"> (  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">f</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">  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a></span><br/>

<a NAME="E2:57_1"/>
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:57_1"/>
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="txt" href="#E1:57_1"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="cfunct_1.html#T71" target="_self" title="CFUNCT_1:th.71">Th71</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="E2:57"/><i><font color="Green" title="E40">A2</font></i>: 
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</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">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a></span><br/>

<a NAME="E3:57"/>
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:57"/>
<font color="Maroon" title="c2">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="txt" href="#E2:57"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="cfunct_1.html#T71" target="_self" title="CFUNCT_1:th.71">Th71</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>
