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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <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>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:77"/><i><font color="Green">E53</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E2:77"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E3:77"/><b>then </b>
 <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:77"><i><font color="Green">E53</font></i></a>, <a class="ref" href="supinf_1.html#T101" target="_self">Th101</a></i>;<br/>
<a NAME="E4:77"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E5:77"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:77"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T91" target="_self">Th91</a></i>;<br/>
<a NAME="E7:77"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:77"/><b>then </b>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T91" target="_self">Th91</a></i>;<br/>
<a NAME="E9:77"/><b>then </b>
 <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:77"><i><font color="Green">E55</font></i></a>, <a class="ref" href="supinf_1.html#T92" target="_self">Th92</a></i>;<br/>
<b>hence </b><a NAME="E10:77"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="supinf_1.html#E4:77"><i><font color="Green">E54</font></i></a>, <a class="ref" href="supinf_1.html#T22" target="_self">Th22</a></i>;<br/>


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