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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:59_1"/><b>then </b>
(  <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp;  <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T40" target="_self">Th40</a>, <a class="ref" href="supinf_2.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E4:59_1"/>
 <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:59"/>
 <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 ;<br/>

<a NAME="E3:59"/><b>then </b>
(  <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp;  <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:59"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T40" target="_self">Th40</a>, <a class="ref" href="supinf_2.html#T41" target="_self">Th41</a></i>;<br/>
<b>hence </b><a NAME="E5:59"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/>


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