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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>   <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  <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>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/> <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> implies  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:3"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T3" title="MEMBERED:th.3">MEMBERED:3</a></span>;<br/>
<a NAME="E3:3"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c3">X</font></span>)</span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c3">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T13" title="RCOMP_1:th.13">RCOMP_1:13</a>, <a class="ref" href="topreal6.html#T68" title="TOPREAL6:th.68">TOPREAL6:68</a></span>;<br/>
<a NAME="E4:3"/>
 <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="measure6.html#T57" title="MEASURE6:th.57">MEASURE6:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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