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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<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>;<br/>





<b>assume </b><a NAME="E1:78"/>
<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">Y</font>
 ;<br/>

<a NAME="E2:78"/><b>then </b><i><font color="Green" title="E43">A1</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">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="c1">Y</font> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">Def11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> or <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1"><b>suppose </b></a><a NAME="E1:78_1_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_1_1"/>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <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">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:78"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T95" target="_self" title="RFUNCT_1:th.95">Th95</a></i>;<br/><b>thus </b><a NAME="E3:78_1_1_1"/>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <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">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:78"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="rfunct_1.html#E1:78_1_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="rfunct_1.html#T96" target="_self" title="RFUNCT_1:th.96">Th96</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_2"><b>suppose </b></a><a NAME="E1:78_1_1_2"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_1_2"/>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <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">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:78"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T96" target="_self" title="RFUNCT_1:th.96">Th96</a></i>;<br/><b>thus </b><a NAME="E3:78_1_1_2"/>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <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">Y</font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E2:78"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="rfunct_1.html#E1:78_1_1_2"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="rfunct_1.html#T95" target="_self" title="RFUNCT_1:th.95">Th95</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:78"/>
<font color="Maroon" title="c2">r</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <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">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D11" target="_self" title="RFUNCT_1:def.11">Def11</a></i>;<br/>


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