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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</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">Vt</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Vt</font> holds <br/>( <font color="Olive" title="b1">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> iff <font color="Olive" title="b2">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c2">V</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Vt</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Vt</font> holds <br/>( <font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> iff <font color="Olive" title="b1">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c3">Vt</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Vt</font> implies ( <font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> iff <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) )</span><br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Vt</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> iff <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">St</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font> implies  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font> )</span><br/><b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c4">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">St</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:22_1_1"/>
<font color="Maroon" title="c5">S</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:22_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="normsp_2.html#T13" target="_self" title="NORMSP_2:th.13">Th13</a></i>;<br/><a NAME="E4:22_1_1"/><b>then </b>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:22_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E1:22_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="nfcont_1.html#D5" title="NFCONT_1:def.5">NFCONT_1:def 5</a></i>;<br/><a NAME="E5:22_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c5">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:22_1_1"/>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:22_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="normsp_2.html#T14" target="_self" title="NORMSP_2:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="frechet.html#D6" title="FRECHET:def.6">FRECHET:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:22"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">Vt</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">S</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">S</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> implies  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E19">A5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">S</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">St</font> = <font color="Maroon" title="c4">S</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/><a NAME="E3:22_2"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c5">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="#E1:22_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="normsp_2.html#T13" target="_self" title="NORMSP_2:th.13">Th13</a></i>;<br/><a NAME="E4:22_2"/><b>then </b>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c5">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:22"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E1:22_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="frechet.html#D6" title="FRECHET:def.6">FRECHET:def 6</a></i>;<br/><a NAME="E5:22_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E3:22_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="normsp_2.html#T14" target="_self" title="NORMSP_2:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E6:22_2"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22"/>
<font color="Maroon" title="c2">V</font> is <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D5" title="NFCONT_1:def.5">NFCONT_1:def 5</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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