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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E2:23"/>
<font color="Maroon">Y</font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a>
 ;<br/>

<a NAME="E3:23"/><b>then </b>
 ex <font color="Olive">UB</font> being   <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">Y</font> st <font color="Olive">UB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">UB</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">UB</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">UB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  )
 ;<br/>
<a NAME="E6:23"/>
( <font color="Maroon">UB</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">UB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D4" target="_self">Def4</a>, , </i>;<br/>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon">X</font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a>
 <b>by </b><i/>;<br/>


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