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<b>let </b><font color="Maroon" title="c1">S</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c1">S</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:1_2_1"/><i><font color="Green" title="E1">Z</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a></span><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S</font> or  not <font color="Maroon" title="c2">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E2:1_2_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E3:1_2_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="#E1:1_2_1"><i><font color="Green" title="E1">Z</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E4:1_2_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="#E1:1_2_1"><i><font color="Green" title="E1">Z</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:1_2_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E1:1_2_1"><i><font color="Green" title="E1">Z</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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