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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</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>assume </b><a NAME="E1:92"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</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">Y</font> )
 ;<br/>

<a NAME="E2:92"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</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> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E3:92"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T42" target="_self">Th42</a>, <a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E4:92"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T43" target="_self">Th43</a>, <a class="ref" href="rfunct_1.html#T39" target="_self">Th39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D11" target="_self">RFUNCT_1:def 11</a><br/>

<b>thus </b><a NAME="E5:92"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T43" target="_self">Th43</a>, <a class="ref" href="rfunct_1.html#T44" target="_self">Th44</a>, <a class="ref" href="rfunct_1.html#T40" target="_self">Th40</a></i>;<br/>


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