<?xml version="1.0"?>
<div class="add">

<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">U</font>, <font color="Maroon" title="c3">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">r</font> be  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:125"/><i><font color="Green" title="E78">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> holds <br/><font color="Olive" title="b1">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font>
 <b>and </b><br/><a NAME="E2:125"/><i><font color="Green" title="E79">A2</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c6">W</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>thus </b><a NAME="E3:125"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c7">F'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E2:125_1"/><i><font color="Green" title="E80">A3</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">a</font>
 <b>and </b><br/><a NAME="E3:125_1"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:125_1"/><i><font color="Green" title="E82">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:125_1"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="rltopsp1.html#E3:125_1"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E6:125_1"/><i><font color="Green" title="E83">A6</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E7:125_1"/><i><font color="Green" title="E84">A7</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:125_1"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="rltopsp1.html#E5:125_1"><i><font color="Green" title="E82">A5</font></i></a></i>;<br/>
<a NAME="E8:125_1"/>
<font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E6:125_1"><i><font color="Green" title="E83">A6</font></i></a></i>;<br/>
<a NAME="E9:125_1"/><b>then </b>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/>
<a NAME="E10:125_1"/><b>then </b><i><font color="Green" title="E85">A8</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E7:125_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:125_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">A</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="E1:125_1_1"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span> 
 ;<br/>

<a NAME="E2:125_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 
;<br/>
<b>hence </b><a NAME="E3:125_1_1"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">F'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> where <font color="Olive" title="b1">a</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E13:125_1"/>
<font color="Maroon" title="c10">F'</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:125_1_2"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F'</font>
 ;<br/>

<a NAME="E2:125_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 
;<br/>
<b>hence </b><a NAME="E3:125_1_2"/>
<font color="Maroon" title="c11">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T50" target="_self" title="RLTOPSP1:th.50">Th50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:125_1"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/>
<a NAME="E15:125_1"/><b>then </b><i><font color="Green" title="E86">A9</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E10:125_1"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E16:125_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:125_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:125_1_3"/><i><font color="Green" title="E87">A10</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font>
 <b>and </b><br/><a NAME="E4:125_1_3"/><i><font color="Green" title="E88">A11</font></i>: 
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a</font> being  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:125_1_3"/><i><font color="Green" title="E89">A12</font></i>: 
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font></span>)</span>
 <b>and </b><br/><a NAME="E7:125_1_3"/><i><font color="Green" title="E90">A13</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:125_1_3"><i><font color="Green" title="E88">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E9:125_1_3"/><i><font color="Green" title="E91">A14</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c14">v</font>
 <b>and </b><br/><a NAME="E10:125_1_3"/><i><font color="Green" title="E92">A15</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:125_1_3"><i><font color="Green" title="E87">A10</font></i></a>, <a class="txt" href="rltopsp1.html#E6:125_1_3"><i><font color="Green" title="E89">A12</font></i></a></i>;<br/>
<a NAME="E11:125_1_3"/><i><font color="Green" title="E93">A16</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E12:125_1_3"/><i><font color="Green" title="E94">A17</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:125"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E7:125_1_3"><i><font color="Green" title="E90">A13</font></i></a></i>;<br/>
<a NAME="E13:125_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:125_1_3"><i><font color="Green" title="E91">A14</font></i></a>, <a class="txt" href="rltopsp1.html#E10:125_1_3"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="rltopsp1.html#E11:125_1_3"><i><font color="Green" title="E93">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:125_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:125_1_3"><i><font color="Green" title="E94">A17</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:125_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<b>hence </b><a NAME="E18:125_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E15:125_1"><i><font color="Green" title="E86">A9</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a><br/>


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

<b>thus </b><a NAME="E4:125"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font> is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" target="_self" title="RLTOPSP1:def.6">RLTOPSP1:def 6</a><br/>

<b>assume </b><a NAME="E1:125_2"/><i><font color="Green" title="E80">A18</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c8">z</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="E2:125_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:125_2"/><i><font color="Green" title="E81">A19</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c9">u</font>
 <b>and </b><br/><a NAME="E5:125_2"/><i><font color="Green" title="E82">A20</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:125_2"/><i><font color="Green" title="E83">A21</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 <b>and </b><br/><a NAME="E8:125_2"/><i><font color="Green" title="E84">A22</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:125_2"><i><font color="Green" title="E82">A20</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:125_2"/><i><font color="Green" title="E85">A23</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font>
 <b>and </b><br/><a NAME="E11:125_2"/><i><font color="Green" title="E86">A24</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:125"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E8:125_2"><i><font color="Green" title="E84">A22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E13:125_2"/><i><font color="Green" title="E87">A25</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c12">v</font>
 <b>and </b><br/><a NAME="E14:125_2"/><i><font color="Green" title="E88">A26</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E7:125_2"><i><font color="Green" title="E83">A21</font></i></a>, <a class="txt" href="rltopsp1.html#E10:125_2"><i><font color="Green" title="E85">A23</font></i></a></i>;<br/>
<a NAME="E15:125_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c12">v</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E4:125_2"><i><font color="Green" title="E81">A19</font></i></a>, <a class="txt" href="rltopsp1.html#E13:125_2"><i><font color="Green" title="E87">A25</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E16:125_2"/><b>then </b><i><font color="Green" title="E89">A27</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E14:125_2"><i><font color="Green" title="E88">A26</font></i></a></i>;<br/>
<a NAME="E17:125_2"/><i><font color="Green" title="E90">A28</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:125_2"><i><font color="Green" title="E80">A18</font></i></a>, <a class="ref" href="real_2.html#T147" title="REAL_2:th.147">REAL_2:147</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_1"/>
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font> or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1_1"><b>suppose </b></a><a NAME="E1:125_2_1_1"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font>
 ;<br/><div class="add"><a NAME="E2:125_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/><a NAME="E3:125_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E4:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E11:125_2"><i><font color="Green" title="E86">A24</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E5:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E17:125_2"><i><font color="Green" title="E90">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:125_2_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E7:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:125"><i><font color="Green" title="E79">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E8:125_2_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E16:125_2"><i><font color="Green" title="E89">A27</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1_2"><b>suppose </b></a><a NAME="E1:125_2_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:125_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/><a NAME="E3:125_2_1_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:125_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:125"><i><font color="Green" title="E79">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E5:125_2_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E16:125_2"><i><font color="Green" title="E89">A27</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c7">z</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="E5:125"/><i><font color="Green" title="E80">A29</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:125"/><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 <b>and </b><br/><a NAME="E9:125"/><i><font color="Green" title="E82">A31</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:125"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E11:125"/><i><font color="Green" title="E83">A32</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font>
 <b>and </b><br/><a NAME="E12:125"/><i><font color="Green" title="E84">A33</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:125"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E9:125"><i><font color="Green" title="E82">A31</font></i></a></i>;<br/>
<a NAME="E13:125"/>
<font color="Maroon" title="c10">a</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:125"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E12:125"><i><font color="Green" title="E84">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:125"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:125"><i><font color="Green" title="E81">A30</font></i></a>, <a class="txt" href="rltopsp1.html#E11:125"><i><font color="Green" title="E83">A32</font></i></a></i>;<br/>


</div>
