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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">T</font> is <a href="frechet.html#V4" title="FRECHET:attr.4">sequential</a> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> ) implies <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a> )</span><br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V4" title="FRECHET:attr.4">sequential</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st not  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font> or <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a> )</span><br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Olive" title="b1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a></span><br/>

<b>assume </b><a NAME="E3:38"/>
not <font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:38"/><i><font color="Green" title="E34">A3</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> holds <br/> not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b2">S</font> ) )
 <b>by </b><i><a class="ref" href="frechet.html#D5" title="FRECHET:def.5">FRECHET:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E7:38"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E8:38"/><i><font color="Green" title="E36">A5</font></i>: 
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">T</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">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">A</font> holds <br/> not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:38"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/>
<a NAME="E9:38"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="frechet2.html#T21" target="_self" title="FRECHET2:th.21">Th21</a></i>;<br/>
<a NAME="E10:38"/>
for <font color="Olive" title="b1">Sq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">Sq</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="Olive" title="b1">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Sq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Sq</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">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> implies  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1"/>
<font color="Maroon" title="c4">Sq</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 <b>and </b><br/><a NAME="E2:38_1"/><i><font color="Green" title="E38">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</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">Sq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">Sq</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E3:38_1"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">Sq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c5">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E5:38_1"/>
<font color="Maroon" title="c6">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:38_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="frechet2.html#E3:38_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E2:38"><i><font color="Green" title="E33">A2</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/>


</div><b>end;</b></div>
<a NAME="E11:38"/><b>then </b>
 <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:38"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="frechet.html#D6" title="FRECHET:def.6">FRECHET:def 6</a></i>;<br/>
<a NAME="E12:38"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E7:38"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="frechet2.html#E9:38"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="pre_topc.html#T45" title="PRE_TOPC:th.45">PRE_TOPC:45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E14:38"/><i><font color="Green" title="E38">A9</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">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/>
<b>thus </b><a NAME="E15:38"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E8:38"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="frechet2.html#E14:38"><i><font color="Green" title="E38">A9</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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