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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#K1">REAL</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:99"/>
<font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:99"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:99"/>
<font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:99"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a><br/>

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E7:99"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="complsp1.html#K21" target="_self">+</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r''</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:99"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">r''</font> &amp; <font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E10:99"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r''</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T47" target="_self">Th47</a>, <a class="ref" href="complsp1.html#T48" target="_self">Th48</a>, <a class="ref" href="complsp1.html#T49" target="_self">Th49</a></i>;<br/>
<b>hence </b><a NAME="E11:99"/>
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T49" target="_self">Th49</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


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