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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">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="seq_1.html#NV1" title="SEQ_1:NV.1">real-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="seq_1.html#NV1" title="SEQ_1:NV.1">real-yielding</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="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font> )</span><br/>





<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E2:85"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">Def11</a></i>;<br/>
<a NAME="E3:85"/><i><font color="Green" title="E50">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:85"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:85"/>
<font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:85"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T102" target="_self" title="RFUNCT_1:th.102">Th102</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span><br/>

<b>thus </b><a NAME="E5:85"/>
<font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:85"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="rfunct_1.html#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T103" target="_self" title="RFUNCT_1:th.103">Th103</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>
