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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">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">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K1">REAL</a> ;<br/>









<b>thus </b><a NAME="E1:99"/>
( <font color="Maroon">f1</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> implies <font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:99_1"/><b>then </b>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f2</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#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E3:99_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f2</font></span>)</span> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E4:99_1"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:99"/>
( <font color="Maroon">f1</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> implies <font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:99_2"/><b>then </b>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f2</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#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E3:99_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f2</font></span>)</span> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E4:99_2"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:99"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f1</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E4:99"/><b>then </b>
 <a href="seq_1.html#K16">-</a> <font color="Maroon">f2</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#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E5:99"/><b>then </b>
<font color="Maroon">f1</font> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <font color="Maroon">f2</font></span>)</span> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E6:99"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>

<a NAME="E7:99"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f2</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#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E8:99"/>
<font color="Maroon">f2</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T38" target="_self">Th38</a></i>;<br/>

<b>thus </b><a NAME="E9:99"/>
<font color="Maroon">f1</font> <a href="seq_1.html#K8">(#)</a> <font color="Maroon">f2</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T46" target="_self">Th46</a>, <a class="ref" href="rfunct_1.html#T47" target="_self">Th47</a>, <a class="ref" href="rfunct_1.html#T38" target="_self">Th38</a></i>;<br/>


</div>
