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<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:97"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:97_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="rfunct_1.html#T42" target="_self">Th42</a></i>;<br/><b>hence </b><a NAME="E2:97_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <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#T44" target="_self">Th44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Olive">r</font> <a href="seq_1.html#K13">(#)</a> <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="E4:97"/>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="rfunct_1.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E5:97"/>
 <a href="seq_1.html#K16">-</a> <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#T44" target="_self">Th44</a></i>;<br/>

<a NAME="E6:97"/>
 <a href="seq_1.html#K21">abs</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="rfunct_1.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E7:97"/>
 <a href="seq_1.html#K21">abs</a> <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#T44" target="_self">Th44</a></i>;<br/>


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