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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1"><b>suppose </b></a><a NAME="E1:106_1_1"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1_1_1"/>
( <font color="Maroon" title="c2">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1_1"><b>suppose </b></a><a NAME="E1:106_1_1_1_1"/>
<font color="Maroon" title="c2">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:106_1_1_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> is <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>
 <b>by </b><i><a class="ref" href="convex1.html#T33" title="CONVEX1:th.33">CONVEX1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1_2"><b>suppose </b></a><a NAME="E1:106_1_1_1_2"/><i><font color="Green" title="E68">A2</font></i>: 
not <font color="Maroon" title="c2">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c4">U</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D12" target="_self" title="RLTOPSP1:def.12">RLTOPSP1:def 12</a><br/><b>take </b><font color="Maroon" title="c5">s</font> = 1;<br/><b>thus </b><a NAME="E2:106_1_1_1_2"/>
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon" title="c6">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E3:106_1_1_1_2"/><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">s</font>
 ;<br/><a NAME="E4:106_1_1_1_2"/><i><font color="Green" title="E70">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:106_1_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E1:106_1_1_1_2"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex1.html#T34" title="CONVEX1:th.34">CONVEX1:34</a></i>;<br/><a NAME="E5:106_1_1_1_2"/>
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:106_1_1_1_2"><i><font color="Green" title="E69">A3</font></i></a></i>;<br/><a NAME="E6:106_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T56" target="_self" title="RLTOPSP1:th.56">Th56</a></i>;<br/><a NAME="E7:106_1_1_1_2"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></i>;<br/><b>hence </b><a NAME="E8:106_1_1_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:106_1_1_1_2"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_2"><b>suppose </b></a><a NAME="E1:106_1_2"/><i><font color="Green" title="E67">A5</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c4">U</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D12" target="_self" title="RLTOPSP1:def.12">RLTOPSP1:def 12</a><br/><a NAME="E2:106_1_2"/>
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:106_1_2"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T50" title="XCMPLX_1:th.50">XCMPLX_1:50</a></i>;<br/><a NAME="E3:106_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T56" target="_self" title="RLTOPSP1:th.56">Th56</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:106_1_2"/><i><font color="Green" title="E68">A6</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:106_1_2"/><i><font color="Green" title="E69">A7</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">s</font> holds <br/><font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D12" target="_self" title="RLTOPSP1:def.12">Def12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/><b>thus </b><a NAME="E7:106_1_2"/>
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:106_1_2"><i><font color="Green" title="E68">A6</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c6">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E8:106_1_2"/><i><font color="Green" title="E70">A8</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">s</font>
 ;<br/><b>let </b><font color="Maroon" title="c7">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="E9:106_1_2"/><i><font color="Green" title="E71">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font>
 ;<br/><a NAME="E10:106_1_2"/>
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:106_1_2"><i><font color="Green" title="E69">A7</font></i></a>, <a class="txt" href="rltopsp1.html#E8:106_1_2"><i><font color="Green" title="E70">A8</font></i></a></i>;<br/><a NAME="E11:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></i>;<br/><a NAME="E12:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></i>;<br/><a NAME="E13:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></i>;<br/><a NAME="E14:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></i>;<br/><a NAME="E15:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:106_1_2"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E16:106_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="ref" href="convex1.html#T32" title="CONVEX1:th.32">CONVEX1:32</a></i>;<br/><b>hence </b><a NAME="E17:106_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E9:106_1_2"><i><font color="Green" title="E71">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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