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<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D12" title="VALUED_0:def.12">VALUED_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c1">f</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> or <span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c1">f</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )</span><br/>

<a NAME="E1:178_1_1"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 
;<br/>
<b>hence </b><a NAME="E2:178_1_1"/>
(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c1">f</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> or <span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c1">f</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )
 <b>by </b><i><a class="ref" href="valued_1.html#D11" target="_self" title="VALUED_1:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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