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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><b>that </b><br/><a NAME="E1:100"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E3:100"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:100"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<a NAME="E5:100"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">X</font> <a href="complsp1.html#K21" target="_self">+</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="complsp1.html#T48" target="_self">Th48</a>, <a class="ref" href="complsp1.html#T50" target="_self">Th50</a>, <a class="ref" href="complsp1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E6:100"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">X</font> <a href="complsp1.html#K21" target="_self">+</a> <font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="complsp1.html#T49" target="_self">Th49</a>, <a class="ref" href="complsp1.html#T51" target="_self">Th51</a>, <a class="ref" href="complsp1.html#T46" target="_self">Th46</a></i>;<br/>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><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="E1:100_1"/>
<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">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:100_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E4:100_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T49" target="_self">Th49</a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E5:100_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T51" target="_self">Th51</a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E6:100_1"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="complsp1.html#T59" target="_self">Th59</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>now </b></a><div class="add"><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="E1:100_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font>
 ;<br/><a NAME="E2:100_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">r'</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<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="E4:100_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="complsp1.html#T48" target="_self">Th48</a>, <a class="ref" href="complsp1.html#T49" target="_self">Th49</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>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="E6:100_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="complsp1.html#T50" target="_self">Th50</a>, <a class="ref" href="complsp1.html#T51" target="_self">Th51</a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font>;<br/><b>thus </b><a NAME="E7:100_2"/>
<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>
 <b>by </b><i><a class="ref" href="complsp1.html#T59" target="_self">Th59</a>, <a class="ref" href="complsp1.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E8:100_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font>
 ;<br/><b>hence </b><a NAME="E9:100_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T59" target="_self">Th59</a>, <a class="ref" href="complsp1.html#T60" target="_self">Th60</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:100"/>
 <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21" target="_self">+</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="complsp1.html#T52" target="_self">Th52</a>, , , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


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