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

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<b>thus </b><a NAME="E1:79"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> implies <font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D9" target="_self">RFUNCT_1:def 9</a><br/>

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:79_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E5:79_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:79_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:79_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:79"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> implies <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D10" target="_self">RFUNCT_1:def 10</a><br/>

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:79_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E5:79_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:79_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:79_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T33" target="_self">Th33</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:79"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> )
 ;<br/>

<b>hence </b><a NAME="E4:79"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>


</div>
