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<span class="kw">let </span><font color="Maroon" title="c1">X</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <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>  &amp; ( for <font color="Olive" title="b2">r9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r9</font> ) holds <br/> <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp; ( for <font color="Olive" title="b1">r9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r9</font> ) implies  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:153"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:153"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
for <font color="Olive" title="b1">r9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font></span><br/>

<a NAME="E3:153"/>
for <font color="Olive" title="b1">r9</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>   st <font color="Olive" title="b1">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E73">A2</font></span></a></span>;<br/>
<a NAME="E4:153"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="153_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">r9</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <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> <font color="Maroon" title="c3">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:153_1"/>
<font color="Maroon" title="c3">r9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <font color="Maroon" title="c3">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font></span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:153_1"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:153_1"/><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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> <font color="Maroon" title="c3">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E4:153"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="seq_4.html#D5" target="_self" title="SEQ_4:def.5">Def5</a></span>;<br/><a NAME="E5:153_1"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="#E3:153_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:153_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> <font color="Maroon" title="c3">r9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:153_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:153"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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