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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:93"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:93"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>and </b><br/><a NAME="E3:93"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E4:93"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c2">Y</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 ;<br/>

<a NAME="E5:93"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:93"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="complsp1.html#E3:93"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="complsp1.html#T94" target="_self" title="COMPLSP1:th.94">Th94</a></i>;<br/>
<a NAME="E6:93"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="complsp1.html#E2:93"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="complsp1.html#E4:93"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="complsp1.html#T95" target="_self" title="COMPLSP1:th.95">Th95</a></i>;<br/>
<div><i><font color="Green" title="E66">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">r1</font>, <font color="Maroon" title="c5">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green" title="E67">A8</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c5">r2</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/><a NAME="E4:93_1"/><i><font color="Green" title="E68">A9</font></i>: 
<font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E2:93"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="complsp1.html#E3:93_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><a NAME="E5:93_1"/>
<font color="Maroon" title="c5">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E4:93"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="complsp1.html#E3:93_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E6:93_1"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E3:93_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="txt" href="complsp1.html#E4:93_1"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">r'</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:93_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">r'</font>
 ;<br/><a NAME="E2:93_2"/><b>then </b><i><font color="Green" title="E67">A10</font></i>: 
<font color="Maroon" title="c3">r'</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:93_2"/><i><font color="Green" title="E68">A11</font></i>: 
( <font color="Maroon" title="c4">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r'</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="complsp1.html#E1:93"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="complsp1.html#E2:93"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">r2</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:93_2"/><i><font color="Green" title="E69">A12</font></i>: 
( <font color="Maroon" title="c5">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r'</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="complsp1.html#E3:93"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="complsp1.html#E4:93"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="complsp1.html#E2:93_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>take </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r2</font>;<br/><b>thus </b><a NAME="E7:93_2"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E4:93_2"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="complsp1.html#E6:93_2"><i><font color="Green" title="E69">A12</font></i></a></i>;<br/><a NAME="E8:93_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r'</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r'</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">r'</font>
 ;<br/><b>hence </b><a NAME="E9:93_2"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">r'</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E4:93_2"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="complsp1.html#E6:93_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:93"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="complsp1.html#E5:93"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="complsp1.html#E6:93"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="complsp1.html#E7:93"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>


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