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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>thus </b><a NAME="E1:59"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E2:59_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:59_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</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>assume </b><a NAME="E2:59"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E3:59"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:59"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a><br/>


</div>
