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

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>  implies <font color="Maroon" title="c1">f</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_1"/>
<font color="Maroon" title="c1">f</font> is  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> </span><br/>

<a NAME="E2:10_3_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:10_3_1"/>
<font color="Maroon" title="c1">f</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D12" onmouseover="rs('valued_0/D12')" onmouseout="rh()">VALUED_0:def 12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
