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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</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">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b2">seq</font> is  <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a>  holds <br/>( <font color="Olive" title="b2">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seq</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <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">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">Y</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b1">seq</font> is  <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a>  holds <br/>( <font color="Olive" title="b1">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <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="c2">Y</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">seq</font> is  <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a>  implies ( <font color="Maroon" title="c3">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq</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="E1:42"/><span class="lab"><font color="Green" title="E23">AS</font></span>: 
( <font color="Maroon" title="c2">Y</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">seq</font> is  <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</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">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:42"/>
<font color="Maroon" title="c3">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D16" title="LOPBAN_1:def.16">LOPBAN_1:def 16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">hence </span><a NAME="E3:42"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq</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="#E1:42"><span class="lab"><font color="Green" title="E23">AS</font></span></a>, <a class="ref" href="nfcont_1.html#D5" title="NFCONT_1:def.5">NFCONT_1:def 5</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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