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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">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">C</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:90"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</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#T106" target="_self" title="RFUNCT_1:th.106">Th106</a>, <a class="ref" href="rfunct_1.html#T108" target="_self" title="RFUNCT_1:th.108">Th108</a></i>;<br/>

<a NAME="E3:90"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:90"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T107" target="_self" title="RFUNCT_1:th.107">Th107</a></i>;<br/>
<b>hence </b><a NAME="E4:90"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</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#T108" target="_self" title="RFUNCT_1:th.108">Th108</a></i>;<br/>

<a NAME="E5:90"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c3">f</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:90"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="rfunct_1.html#T107" target="_self" title="RFUNCT_1:th.107">Th107</a></i>;<br/>
<b>hence </b><a NAME="E6:90"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</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#T108" target="_self" title="RFUNCT_1:th.108">Th108</a></i>;<br/>


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