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

<b>let </b><font color="Maroon" title="c1">Y</font> be    <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="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> holds <br/>( <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies ( <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )</span><br/>



<a NAME="E1:94"/><i><font color="Green" title="E44">F</font></i>: 
<span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T62" target="_self" title="RFUNCT_1:th.62">Th62</a></i>;<br/>
<b>assume </b><a NAME="E2:94"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )</span><br/>

<b>hence </b><a NAME="E3:94"/>
<span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E44">F</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a></span><br/>

<b>thus </b><a NAME="E4:94"/>
<span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Y</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="#E2:94"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T97" target="_self" title="RFUNCT_1:th.97">Th97</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>
