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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>thus </b><a NAME="E1:36"/>
<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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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><br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1"/>
( <font color="Maroon" title="c4">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> or <font color="Maroon" title="c4">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="c3">B</font> )
 </span><b>by </b><i><a class="txt" href="frechet2.html#E1:36_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>consider </b><font color="Maroon" title="c6">S</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:36_1_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E5:36_1_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c5">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="c6">S</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:36_1_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/><a NAME="E6:36_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:36_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4:36_1_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E8:36_1_1_1"/>
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:36_1_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c4">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="c3">B</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>consider </b><font color="Maroon" title="c6">S</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:36_1_1_2"/><i><font color="Green" title="E32">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E5:36_1_1_2"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c5">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="c6">S</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:36_1_1_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/><a NAME="E6:36_1_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:36_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E4:36_1_1_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E8:36_1_1_2"/>
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:36_1_1_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:36"/>
 <a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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><br/>

<b>assume </b><a NAME="E1:36_2"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">S</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:36_2"/><i><font color="Green" title="E31">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E5:36_2"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c5">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="c6">S</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:36_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="frechet2.html#D2" target="_self" title="FRECHET2:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">S1</font> being    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c6">S</font><b> such that </b><br/><a NAME="E7:36_2"/><i><font color="Green" title="E33">A11</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> or  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="frechet2.html#E4:36_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="frechet2.html#T7" target="_self" title="FRECHET2:th.7">Th7</a></i>;<br/>
<a NAME="E8:36_2"/><i><font color="Green" title="E34">A12</font></i>: 
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c7">S1</font>
 
<b>by </b><i><a class="ref" href="frechet2.html#T19" target="_self" title="FRECHET2:th.19">Th19</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> or  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 </span><b>by </b><i><a class="txt" href="frechet2.html#E7:36_2"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><a NAME="E2:36_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">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#E5:36_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="frechet2.html#E8:36_2"><i><font color="Green" title="E34">A12</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="E3:36_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><a NAME="E2:36_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">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="c3">B</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5:36_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="frechet2.html#E8:36_2"><i><font color="Green" title="E34">A12</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="E3:36_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="frechet2.html#K3" title="FRECHET2:func.3">Cl_Seq</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
