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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>









<b>thus </b><a NAME="E1:98"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:98_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T108" target="_self">Th108</a></i>;<br/>
<a NAME="E3:98_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:98_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:98_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rfunct_1.html#T100" target="_self">Th100</a></i>;<br/>


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

<b>thus </b><a NAME="E2:98"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:98_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T108" target="_self">Th108</a></i>;<br/>
<a NAME="E3:98_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:98_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:98_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rfunct_1.html#T100" target="_self">Th100</a></i>;<br/>


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

<b>assume </b><a NAME="E3:98"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:98"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T108" target="_self">Th108</a></i>;<br/>
<b>hence </b><a NAME="E5:98"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E3:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rfunct_1.html#T100" target="_self">Th100</a></i>;<br/>


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