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<b>let </b><font color="Maroon">X</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> , <font color="Maroon">Y</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>assume </b><a NAME="E1:63"/><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>set </b><font color="Maroon">S2</font> =  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E2:63"/>
(  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">Y</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">Y</font> &amp; ( for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">Y</font> holds <br/> <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">Y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:63"/><b>then </b>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">Y</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T36" target="_self">Th36</a>, </i>;<br/>
<b>hence </b><a NAME="E4:63"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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