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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">V</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">St</font> being   <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>  st <font color="Olive" title="b1">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font> holds <br/> <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Olive" title="b1">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Vt</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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#K2" title="FRECHET:func.2">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/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c4">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:22_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="frechet.html#K2" title="FRECHET:func.2">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/><span class="kw">reconsider </span><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="E4:22_1_1"/>
<font color="Maroon" title="c5">S</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="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="normsp_2.html#T13" target="_self" onmouseover="rs('normsp_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E5:22_1_1"/><span class="kw">then </span>
 <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="nfcont_1.html#D3" onmouseover="rs('nfcont_1/D3')" onmouseout="rh()">NFCONT_1:def 3</a></span>;<br/><a NAME="E6:22_1_1"/><span class="kw">then </span>
<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#K5" title="NORMSP_1:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E7:22_1_1"/>
 <a href="frechet.html#K2" title="FRECHET:func.2">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="normsp_2.html#T14" target="_self" onmouseover="rs('normsp_2/T14')" onmouseout="rh()">Th14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D7" onmouseover="rs('frechet/D7')" onmouseout="rh()">FRECHET:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:22"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c3">Vt</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  holds <br/> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E2:22_2"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/><span class="kw">reconsider </span><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="E4:22_2"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c5">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="normsp_2.html#T13" target="_self" onmouseover="rs('normsp_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E5:22_2"/><span class="kw">then </span>
 <a href="frechet.html#K2" title="FRECHET:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="frechet.html#D7" onmouseover="rs('frechet/D7')" onmouseout="rh()">FRECHET:def 7</a></span>;<br/><a NAME="E6:22_2"/><span class="kw">then </span>
<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#K5" title="NORMSP_1:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="normsp_2.html#T14" target="_self" onmouseover="rs('normsp_2/T14')" onmouseout="rh()">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:22_2"/>
 <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#D3" onmouseover="rs('nfcont_1/D3')" onmouseout="rh()">NFCONT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
