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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">K</font> be  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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">C</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1"/>
( <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1"><b>suppose </b></a><a NAME="E1:109_1_1"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">V</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>;<br/><b>thus </b><a NAME="E2:109_1_1"/>
<font color="Maroon" title="c4">V</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="topgrp_1.html#T21" title="TOPGRP_1:th.21">TOPGRP_1:21</a></i>;<br/><a NAME="E3:109_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="convex1.html#T40" title="CONVEX1:th.40">CONVEX1:40</a></i>;<br/><b>hence </b><a NAME="E4:109_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c4">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_2"><b>suppose </b></a><a NAME="E1:109_1_2"/><i><font color="Green" title="E70">A3</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green" title="E71">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:109_1_2_1"/><i><font color="Green" title="E72">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 ;<br/><a NAME="E2:109_1_2_1"/><i><font color="Green" title="E73">A6</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D9" title="RUSUB_4:def.9">RUSUB_4:def 9</a></i>;<br/><a NAME="E3:109_1_2_1"/>
<font color="Maroon" title="c2">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E4:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="txt" href="rltopsp1.html#E1:109_1_2_1"><i><font color="Green" title="E72">A5</font></i></a></i>;<br/><a NAME="E5:109_1_2_1"/><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> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>;<br/><a NAME="E6:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> 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="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">V'</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E8:109_1_2_1"/>
<font color="Maroon" title="c5">V'</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>and </b><br/><a NAME="E9:109_1_2_1"/><i><font color="Green" title="E74">A7</font></i>: 
<font color="Maroon" title="c5">V'</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">V'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T57" target="_self" title="RLTOPSP1:th.57">Th57</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E11:109_1_2_1"/><i><font color="Green" title="E75">A8</font></i>: 
<font color="Maroon" title="c6">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>and </b><br/><a NAME="E12:109_1_2_1"/><i><font color="Green" title="E76">A9</font></i>: 
<font color="Maroon" title="c6">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c6">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V'</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T57" target="_self" title="RLTOPSP1:th.57">Th57</a></i>;<br/><b>take </b><font color="Maroon" title="c7">Vx</font> = <font color="Maroon" title="c6">Vx</font>;<br/><b>thus </b><a NAME="E13:109_1_2_1"/>
<font color="Maroon" title="c7">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E11:109_1_2_1"><i><font color="Green" title="E75">A8</font></i></a></i>;<br/><a NAME="E14:109_1_2_1"/>
<font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_1"><b>proof </b></a><div class="add">

<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="E1:109_1_2_1_1"/><i><font color="Green" title="E77">A10</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Vx</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">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/>
<a NAME="E3:109_1_2_1_1"/>
 <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="c7">Vx</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></i>;<br/>
<a NAME="E4:109_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2_1_1"><i><font color="Green" title="E77">A10</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self" title="RLTOPSP1:th.3">Th3</a></i>;<br/>
<a NAME="E5:109_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:109_1_2_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V'</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2_1"><i><font color="Green" title="E76">A9</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:109_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V'</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">V'</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2_1"><i><font color="Green" title="E76">A9</font></i></a>, <a class="ref" href="rltopsp1.html#T11" target="_self" title="RLTOPSP1:th.11">Th11</a></i>;<br/><a NAME="E16:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2_1"><i><font color="Green" title="E74">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:109_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T8" target="_self" title="RLTOPSP1:th.8">Th8</a></i>;<br/><a NAME="E18:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Vx</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T7" target="_self" title="RLTOPSP1:th.7">Th7</a></i>;<br/><a NAME="E19:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T36" title="CONVEX1:th.36">CONVEX1:36</a></i>;<br/><a NAME="E20:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T6" target="_self" title="RLTOPSP1:th.6">Th6</a></i>;<br/><a NAME="E21:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<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> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></i>;<br/><a NAME="E22:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#T5" target="_self" title="RLTOPSP1:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E23:109_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c4">xV</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:109_1_2_2"/><i><font color="Green" title="E72">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_2"><i><font color="Green" title="E72">A11</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E5:109_1_2_2"/><i><font color="Green" title="E73">A12</font></i>: 
<font color="Maroon" title="c7">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>and </b><br/><a NAME="E6:109_1_2_2"/><i><font color="Green" title="E74">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c7">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="rltopsp1.html#E2:109_1_2_2"><i><font color="Green" title="E72">A11</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">z</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/><b>thus </b><a NAME="E7:109_1_2_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_2"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_2"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_2"><i><font color="Green" title="E74">A13</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">xV</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">V1</font>, <font color="Olive" title="b4">V2</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">V1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">v2</font>,<font color="Olive" title="b4">V2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">v1</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b4">V2</font> );<br/><a NAME="E5:109_1_2"/><i><font color="Green" title="E72">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:109_1_2_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 ;<br/>

<a NAME="E2:109_1_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">v</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">V</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">V</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">V</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">V</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font>]
 ;<br/>


</div><b>end;</b></div><a NAME="E6:109_1_2"/><i><font color="Green" title="E73">A15</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 ;<br/><a NAME="E7:109_1_2"/><i><font color="Green" title="E74">A16</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>given </b><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c11">V1</font>, <font color="Maroon" title="c12">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><a NAME="E2:109_1_2_4"/><i><font color="Green" title="E75">A17</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c11">V1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E3:109_1_2_4"/><i><font color="Green" title="E76">A18</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c12">V2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:109_1_2_4"/><i><font color="Green" title="E77">A19</font></i>: 
<font color="Maroon" title="c9">v1</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c11">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v2</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c12">V2</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c13">w1</font>, <font color="Maroon" title="c14">w2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c15">W1</font>, <font color="Maroon" title="c16">W2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><a NAME="E6:109_1_2_4"/><i><font color="Green" title="E78">A20</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">w1</font>,<font color="Maroon" title="c15">W1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E7:109_1_2_4"/><i><font color="Green" title="E79">A21</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c16">W2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E8:109_1_2_4"/><i><font color="Green" title="E80">A22</font></i>: 
<font color="Maroon" title="c13">w1</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c15">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w2</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c16">W2</font>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c9">v1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">w2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">V1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">W2</font>
;<br/>

<a NAME="E9:109_1_2_4"/>
( <font color="Maroon" title="c10">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w1</font> &amp; <font color="Maroon" title="c12">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">W1</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_4"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_4"><i><font color="Green" title="E78">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E10:109_1_2_4"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c11">V1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c16">W2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">v1</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c11">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w2</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c16">W2</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_4"><i><font color="Green" title="E75">A17</font></i></a>, <a class="txt" href="rltopsp1.html#E4:109_1_2_4"><i><font color="Green" title="E77">A19</font></i></a>, <a class="txt" href="rltopsp1.html#E7:109_1_2_4"><i><font color="Green" title="E79">A21</font></i></a>, <a class="txt" href="rltopsp1.html#E8:109_1_2_4"><i><font color="Green" title="E80">A22</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c6">eqR</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c5">xV</font><b> such that </b><br/><a NAME="E9:109_1_2"/><i><font color="Green" title="E75">A23</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">eqR</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1" title="EQREL_1:sch.1">EQREL_1:sch 1</a>(<a class="txt" href="rltopsp1.html#E5:109_1_2"><i><font color="Green" title="E72">A14</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2"><i><font color="Green" title="E73">A15</font></i></a>, <a class="txt" href="rltopsp1.html#E7:109_1_2"><i><font color="Green" title="E74">A16</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:109_1_2_5"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">eqR</font>
 ;<br/><a NAME="E2:109_1_2_5"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:109_1_2_5"/>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:109_1_2"/><i><font color="Green" title="E76">A24</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">eqR</font>
 <b>and </b><br/><a NAME="E13:109_1_2"/><i><font color="Green" title="E77">A25</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">eqR</font> holds <br/><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28" title="WELLORD2:th.28">WELLORD2:28</a></i>;<br/><a NAME="E14:109_1_2"/><i><font color="Green" title="E78">A26</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">xV</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:109_1_2_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 ;<br/>

<b>then 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="E3:109_1_2_6"/><i><font color="Green" title="E79">A27</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:109_1_2_6"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">xV</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="rltopsp1.html#E3:109_1_2_6"><i><font color="Green" title="E79">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:109_1_2_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 ;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c8">xVV</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>;<br/><b>set </b><font color="Maroon" title="c9">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> </span>}</span> ;<br/><a NAME="E15:109_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</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="109_1_2_7"><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="E1:109_1_2_7"/>
<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">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_7"/><b>then </b>
 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">X</font> ex <font color="Olive" title="b2">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_7"/>
<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">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</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="E17:109_1_2"/><i><font color="Green" title="E79">A28</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_8"><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>,<a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a><br/>

<b>assume </b><a NAME="E1:109_1_2_8"/><i><font color="Green" title="E80">A29</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</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="c13">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E4:109_1_2_8"/><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c13">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>and </b><br/><a NAME="E5:109_1_2_8"/><i><font color="Green" title="E82">A31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c13">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c13">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2"><i><font color="Green" title="E71">A4</font></i></a>, <a class="txt" href="rltopsp1.html#E1:109_1_2_8"><i><font color="Green" title="E80">A29</font></i></a></i>;<br/>
<a NAME="E6:109_1_2_8"/>
 <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="c13">Vx</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></i>;<br/>
<a NAME="E7:109_1_2_8"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E4"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<a NAME="E8:109_1_2_8"/><b>then </b><i><font color="Green" title="E83">A32</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<a NAME="E9:109_1_2_8"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2_8"><i><font color="Green" title="E80">A29</font></i></a>, <a class="txt" href="rltopsp1.html#E4:109_1_2_8"><i><font color="Green" title="E81">A30</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_8"><i><font color="Green" title="E82">A31</font></i></a></i>;<br/>
<a NAME="E10:109_1_2_8"/><b>then </b><i><font color="Green" title="E84">A33</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">eqR</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E11:109_1_2_8"/><b>then </b>
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a></i>;<br/>
<a NAME="E12:109_1_2_8"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">eqR</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c15">v2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c16">V1</font>, <font color="Maroon" title="c17">V2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E14:109_1_2_8"/><i><font color="Green" title="E85">A34</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c16">V1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E15:109_1_2_8"/><i><font color="Green" title="E86">A35</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">v2</font>,<font color="Maroon" title="c17">V2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E16:109_1_2_8"/><i><font color="Green" title="E87">A36</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c16">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v2</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">V2</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2"><i><font color="Green" title="E75">A23</font></i></a></i>;<br/>
<a NAME="E17:109_1_2_8"/><i><font color="Green" title="E88">A37</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v2</font> &amp; <font color="Maroon" title="c13">Vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">V2</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E15:109_1_2_8"><i><font color="Green" title="E86">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:109_1_2_8"/>
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E10:109_1_2_8"><i><font color="Green" title="E84">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:109_1_2_8"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</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="txt" href="rltopsp1.html#E14:109_1_2_8"><i><font color="Green" title="E85">A34</font></i></a>, <a class="txt" href="rltopsp1.html#E16:109_1_2_8"><i><font color="Green" title="E87">A36</font></i></a>, <a class="txt" href="rltopsp1.html#E17:109_1_2_8"><i><font color="Green" title="E88">A37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:109_1_2_8"/>
<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>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_8"><i><font color="Green" title="E83">A32</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="E18:109_1_2"/>
<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="109_1_2_9"><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="E1:109_1_2_9"/>
<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="E2:109_1_2_9"/><b>then </b>
 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">X</font> ex <font color="Olive" title="b2">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_9"/>
<font color="Maroon" title="c11">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E20:109_1_2"/><i><font color="Green" title="E80">A38</font></i>: 
<font color="Maroon" title="c11">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F</font>
 <b>and </b><br/><a NAME="E21:109_1_2"/><i><font color="Green" title="E81">A39</font></i>: 
<font color="Maroon" title="c11">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E22:109_1_2"/><i><font color="Green" title="E82">A40</font></i>: 
<font color="Maroon" title="c11">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E17:109_1_2"><i><font color="Green" title="E79">A28</font></i></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></i>;<br/><a NAME="E23:109_1_2"/><i><font color="Green" title="E83">A41</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">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b3">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:109_1_2_10"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 ;<br/>

<a NAME="E2:109_1_2_10"/><b>then </b>
<font color="Maroon" title="c12">A</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="txt" href="rltopsp1.html#E20:109_1_2"><i><font color="Green" title="E80">A38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:109_1_2_10"/>
 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">X</font> ex <font color="Olive" title="b2">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E24:109_1_2"/><i><font color="Green" title="E84">A42</font></i>: 
for <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">X</font><br/> for <font color="Olive" title="b2">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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>  st <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c13">Vx</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:109_1_2_11"/><i><font color="Green" title="E85">A43</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>and </b><br/><a NAME="E2:109_1_2_11"/><i><font color="Green" title="E86">A44</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 ;<br/>

<a NAME="E3:109_1_2_11"/>
<font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</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="txt" href="rltopsp1.html#E20:109_1_2"><i><font color="Green" title="E80">A38</font></i></a>, <a class="txt" href="rltopsp1.html#E1:109_1_2_11"><i><font color="Green" title="E85">A43</font></i></a></i>;<br/>
<b>then 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>, <font color="Maroon" title="c15">Vv</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E5:109_1_2_11"/><i><font color="Green" title="E87">A45</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c15">Vv</font>
 <b>and </b><br/><a NAME="E6:109_1_2_11"/><i><font color="Green" title="E88">A46</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 ;<br/>
<a NAME="E7:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E14:109_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_11"><i><font color="Green" title="E88">A46</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c17">Vu</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E9:109_1_2_11"/><i><font color="Green" title="E89">A47</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c17">Vu</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E10:109_1_2_11"/><i><font color="Green" title="E90">A48</font></i>: 
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c17">Vu</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vu</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c17">Vu</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c17">Vu</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 ;<br/>
<a NAME="E11:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">eqR</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_11"><i><font color="Green" title="E87">A45</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_11"><i><font color="Green" title="E88">A46</font></i></a>, <a class="txt" href="rltopsp1.html#E14:109_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="rltopsp1.html#E2:109_1_2_11"><i><font color="Green" title="E86">A44</font></i></a></i>;<br/>
<a NAME="E12:109_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E13:109_1_2_11"/><b>then </b><i><font color="Green" title="E91">A49</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E14:109_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_11"><i><font color="Green" title="E88">A46</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:109_1_2_11"/><i><font color="Green" title="E92">A50</font></i>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E16:109_1_2_11"/><i><font color="Green" title="E93">A51</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">A</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_11"><i><font color="Green" title="E86">A44</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:109_1_2_11"/><i><font color="Green" title="E94">A52</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E19:109_1_2_11"/><i><font color="Green" title="E95">A53</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">B</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:109_1_2_11"><i><font color="Green" title="E88">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:109_1_2_11"/><i><font color="Green" title="E96">A54</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 <b>and </b><br/><a NAME="E22:109_1_2_11"/><i><font color="Green" title="E97">A55</font></i>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c20">a</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E15:109_1_2_11"><i><font color="Green" title="E92">A50</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:109_1_2_11"/><i><font color="Green" title="E98">A56</font></i>: 
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 <b>and </b><br/><a NAME="E25:109_1_2_11"/><i><font color="Green" title="E99">A57</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c21">b</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E18:109_1_2_11"><i><font color="Green" title="E94">A52</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E26:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c20">a</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="rltopsp1.html#E15:109_1_2_11"><i><font color="Green" title="E92">A50</font></i></a>, <a class="txt" href="rltopsp1.html#E16:109_1_2_11"><i><font color="Green" title="E93">A51</font></i></a>, <a class="txt" href="rltopsp1.html#E22:109_1_2_11"><i><font color="Green" title="E97">A55</font></i></a></i>;<br/>
<a NAME="E27:109_1_2_11"/><b>then </b><i><font color="Green" title="E100">A58</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E21:109_1_2_11"><i><font color="Green" title="E96">A54</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E28:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c21">b</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="rltopsp1.html#E18:109_1_2_11"><i><font color="Green" title="E94">A52</font></i></a>, <a class="txt" href="rltopsp1.html#E19:109_1_2_11"><i><font color="Green" title="E95">A53</font></i></a>, <a class="txt" href="rltopsp1.html#E25:109_1_2_11"><i><font color="Green" title="E99">A57</font></i></a></i>;<br/>
<a NAME="E29:109_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E13:109_1_2_11"><i><font color="Green" title="E91">A49</font></i></a>, <a class="txt" href="rltopsp1.html#E16:109_1_2_11"><i><font color="Green" title="E93">A51</font></i></a>, <a class="txt" href="rltopsp1.html#E19:109_1_2_11"><i><font color="Green" title="E95">A53</font></i></a>, <a class="txt" href="rltopsp1.html#E22:109_1_2_11"><i><font color="Green" title="E97">A55</font></i></a>, <a class="txt" href="rltopsp1.html#E25:109_1_2_11"><i><font color="Green" title="E99">A57</font></i></a>, <a class="txt" href="rltopsp1.html#E27:109_1_2_11"><i><font color="Green" title="E100">A58</font></i></a>, <a class="txt" href="rltopsp1.html#E24:109_1_2_11"><i><font color="Green" title="E98">A56</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E30:109_1_2_11"/><b>then </b>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v</font> &amp; <font color="Maroon" title="c13">Vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Vv</font> &amp; <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c15">Vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Vu</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2_11"><i><font color="Green" title="E89">A47</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E31:109_1_2_11"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c13">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c13">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c13">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c13">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E10:109_1_2_11"><i><font color="Green" title="E90">A48</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c12">FVx</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Vx</font> where <font color="Olive" title="b1">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> :  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> ;<br/><a NAME="E25:109_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">Vx</font> where <font color="Olive" title="b1">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> :  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</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="109_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="E1:109_1_2_12"/>
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Vx</font> where <font color="Olive" title="b1">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> :  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_12"/><b>then </b>
 ex <font color="Olive" title="b1">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Vx</font> &amp;  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_12"/>
<font color="Maroon" title="c13">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="c13">FVx</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Vx</font> where <font color="Olive" title="b1">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> :  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</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/><b>take </b><font color="Maroon" title="c14">V</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c13">FVx</font>;<br/><a NAME="E27:109_1_2"/><i><font color="Green" title="E85">A59</font></i>: 
<font color="Maroon" title="c13">FVx</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="109_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="E1:109_1_2_13"/>
<font color="Maroon" title="c15">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 ;<br/>

<a NAME="E2:109_1_2_13"/><b>then </b>
 ex <font color="Olive" title="b1">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c15">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Vx</font> &amp;  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_13"/>
<font color="Maroon" title="c15">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E86">A60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_14"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c15">y</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="E2:109_1_2_14"/><i><font color="Green" title="E87">A61</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:109_1_2_14"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W</font>
 <b>and </b><br/><a NAME="E5:109_1_2_14"/><i><font color="Green" title="E88">A62</font></i>: 
<font color="Maroon" title="c16">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21:109_1_2"><i><font color="Green" title="E81">A39</font></i></a>, <a class="txt" href="rltopsp1.html#E2:109_1_2_14"><i><font color="Green" title="E87">A61</font></i></a>, <a class="ref" href="rltopsp1.html#T2" target="_self" title="RLTOPSP1:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c18">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E7:109_1_2_14"/><i><font color="Green" title="E89">A63</font></i>: 
( <font color="Maroon" title="c16">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c18">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E23:109_1_2"><i><font color="Green" title="E83">A41</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_14"><i><font color="Green" title="E88">A62</font></i></a></i>;<br/><a NAME="E8:109_1_2_14"/>
<font color="Maroon" title="c18">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:109_1_2_14"><i><font color="Green" title="E88">A62</font></i></a>, <a class="txt" href="rltopsp1.html#E7:109_1_2_14"><i><font color="Green" title="E89">A63</font></i></a></i>;<br/><b>hence </b><a NAME="E9:109_1_2_14"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:109_1_2_15"/><i><font color="Green" title="E87">A64</font></i>: 
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">A'</font> = <font color="Maroon" title="c15">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:109_1_2_15"/><i><font color="Green" title="E88">A65</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c16">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">A'</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="E4:109_1_2_15"/>
 ex <font color="Olive" title="b1">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Vx</font> &amp;  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b1">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2_15"><i><font color="Green" title="E87">A64</font></i></a></i>;<br/><a NAME="E5:109_1_2_15"/><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>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c16">A'</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/><b>hence </b><a NAME="E6:109_1_2_15"/>
 <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="c15">A</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_15"><i><font color="Green" title="E88">A65</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E30:109_1_2"/><b>then </b><i><font color="Green" title="E87">A66</font></i>: 
 <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="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E28:109_1_2"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> 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">X</font> ex <font color="Olive" title="b2">Vx</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> &amp; <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Vx</font> );<br/><a NAME="E31:109_1_2"/><i><font color="Green" title="E88">A67</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:109_1_2_16"/><i><font color="Green" title="E89">A68</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 ;<br/>

<a NAME="E2:109_1_2_16"/><b>then </b>
<font color="Maroon" title="c15">x</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="txt" href="rltopsp1.html#E20:109_1_2"><i><font color="Green" title="E80">A38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c17">Vz</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E4:109_1_2_16"/><i><font color="Green" title="E90">A69</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vz</font>
 <b>and </b><br/><a NAME="E5:109_1_2_16"/><i><font color="Green" title="E91">A70</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c17">Vz</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c17">Vz</font>
;<br/>

<b>thus </b><a NAME="E6:109_1_2_16"/>
( <font color="Maroon" title="c17">Vz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">Vz</font>] )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2_16"><i><font color="Green" title="E89">A68</font></i></a>, <a class="txt" href="rltopsp1.html#E4:109_1_2_16"><i><font color="Green" title="E90">A69</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_16"><i><font color="Green" title="E91">A70</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c15">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">G</font>,<font color="Maroon" title="c13">FVx</font><b> such that </b><br/><a NAME="E33:109_1_2"/><i><font color="Green" title="E89">A71</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="rltopsp1.html#E31:109_1_2"><i><font color="Green" title="E88">A67</font></i></a>);<br/></i><a NAME="E34:109_1_2"/><i><font color="Green" title="E90">A72</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E28:109_1_2"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:109_1_2"/><i><font color="Green" title="E91">A73</font></i>: 
<font color="Maroon" title="c13">FVx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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="E1:109_1_2_17"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E3:109_1_2_17"/><i><font color="Green" title="E92">A74</font></i>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Vx</font>
 <b>and </b><br/><a NAME="E4:109_1_2_17"/><i><font color="Green" title="E93">A75</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">X</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">x</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="E6:109_1_2_17"/><i><font color="Green" title="E94">A76</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>and </b><br/><a NAME="E7:109_1_2_17"/><i><font color="Green" title="E95">A77</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:109_1_2_17"><i><font color="Green" title="E93">A75</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c20">Vv</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E9:109_1_2_17"/><i><font color="Green" title="E96">A78</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">Vv</font>
 <b>and </b><br/><a NAME="E10:109_1_2_17"/>
<font color="Maroon" title="c19">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">Vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>and </b><br/><a NAME="E11:109_1_2_17"/><i><font color="Green" title="E97">A79</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E12:109_1_2_17"/><i><font color="Green" title="E98">A80</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c17">Vx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Vv</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E33:109_1_2"><i><font color="Green" title="E89">A71</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_17"><i><font color="Green" title="E94">A76</font></i></a></i>;<br/>
<a NAME="E13:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">eqR</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_17"><i><font color="Green" title="E96">A78</font></i></a>, <a class="txt" href="rltopsp1.html#E14:109_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="rltopsp1.html#E7:109_1_2_17"><i><font color="Green" title="E95">A77</font></i></a>, <a class="txt" href="rltopsp1.html#E11:109_1_2_17"><i><font color="Green" title="E97">A79</font></i></a></i>;<br/>
<a NAME="E14:109_1_2_17"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E15:109_1_2_17"/><b>then </b><i><font color="Green" title="E99">A81</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E14:109_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="rltopsp1.html#E11:109_1_2_17"><i><font color="Green" title="E97">A79</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:109_1_2_17"/><i><font color="Green" title="E100">A82</font></i>: 
<font color="Maroon" title="c21">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E18:109_1_2_17"/><i><font color="Green" title="E101">A83</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">A</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E7:109_1_2_17"><i><font color="Green" title="E95">A77</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:109_1_2_17"/><i><font color="Green" title="E102">A84</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E21:109_1_2_17"/><i><font color="Green" title="E103">A85</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E11:109_1_2_17"><i><font color="Green" title="E97">A79</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:109_1_2_17"/><i><font color="Green" title="E104">A86</font></i>: 
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 <b>and </b><br/><a NAME="E24:109_1_2_17"/><i><font color="Green" title="E105">A87</font></i>: 
<font color="Maroon" title="c21">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c23">a</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E17:109_1_2_17"><i><font color="Green" title="E100">A82</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E26:109_1_2_17"/><i><font color="Green" title="E106">A88</font></i>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">xV</font>
 <b>and </b><br/><a NAME="E27:109_1_2_17"/><i><font color="Green" title="E107">A89</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c24">b</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E20:109_1_2_17"><i><font color="Green" title="E102">A84</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E28:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c23">a</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="rltopsp1.html#E17:109_1_2_17"><i><font color="Green" title="E100">A82</font></i></a>, <a class="txt" href="rltopsp1.html#E18:109_1_2_17"><i><font color="Green" title="E101">A83</font></i></a>, <a class="txt" href="rltopsp1.html#E24:109_1_2_17"><i><font color="Green" title="E105">A87</font></i></a></i>;<br/>
<a NAME="E29:109_1_2_17"/><b>then </b><i><font color="Green" title="E108">A90</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E23:109_1_2_17"><i><font color="Green" title="E104">A86</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E30:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">eqR</font>,<font color="Maroon" title="c24">b</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="rltopsp1.html#E13:109_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="rltopsp1.html#E20:109_1_2_17"><i><font color="Green" title="E102">A84</font></i></a>, <a class="txt" href="rltopsp1.html#E21:109_1_2_17"><i><font color="Green" title="E103">A85</font></i></a>, <a class="txt" href="rltopsp1.html#E27:109_1_2_17"><i><font color="Green" title="E107">A89</font></i></a></i>;<br/>
<a NAME="E31:109_1_2_17"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">Vv</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E15:109_1_2_17"><i><font color="Green" title="E99">A81</font></i></a>, <a class="txt" href="rltopsp1.html#E18:109_1_2_17"><i><font color="Green" title="E101">A83</font></i></a>, <a class="txt" href="rltopsp1.html#E21:109_1_2_17"><i><font color="Green" title="E103">A85</font></i></a>, <a class="txt" href="rltopsp1.html#E24:109_1_2_17"><i><font color="Green" title="E105">A87</font></i></a>, <a class="txt" href="rltopsp1.html#E27:109_1_2_17"><i><font color="Green" title="E107">A89</font></i></a>, <a class="txt" href="rltopsp1.html#E29:109_1_2_17"><i><font color="Green" title="E108">A90</font></i></a>, <a class="txt" href="rltopsp1.html#E26:109_1_2_17"><i><font color="Green" title="E106">A88</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E32:109_1_2_17"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font> &amp; <font color="Maroon" title="c17">Vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Vv</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E33:109_1_2_17"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E34:109_1_2"><i><font color="Green" title="E90">A72</font></i></a>, <a class="txt" href="rltopsp1.html#E3:109_1_2_17"><i><font color="Green" title="E92">A74</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_17"><i><font color="Green" title="E94">A76</font></i></a>, <a class="txt" href="rltopsp1.html#E12:109_1_2_17"><i><font color="Green" title="E98">A80</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:109_1_2"/>
<font color="Maroon" title="c15">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c13">FVx</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E22:109_1_2"><i><font color="Green" title="E82">A40</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E37:109_1_2"/><b>then </b>
<font color="Maroon" title="c13">FVx</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E35:109_1_2"><i><font color="Green" title="E91">A73</font></i></a>, <a class="ref" href="finset_1.html#T27" title="FINSET_1:th.27">FINSET_1:27</a></i>;<br/><a NAME="E38:109_1_2"/><b>then </b>
<font color="Maroon" title="c14">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E27:109_1_2"><i><font color="Green" title="E85">A59</font></i></a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></i>;<br/><a NAME="E39:109_1_2"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c14">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><b>hence </b><a NAME="E40:109_1_2"/>
<font color="Maroon" title="c14">V</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="txt" href="rltopsp1.html#E30:109_1_2"><i><font color="Green" title="E87">A66</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c16">FxVxV</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c17">FxVxVx</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> ;<br/><a NAME="E41:109_1_2"/><i><font color="Green" title="E92">A91</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:109_1_2_18"/><i><font color="Green" title="E93">A92</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font>
 ;<br/>

<a NAME="E2:109_1_2_18"/>
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</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">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c20">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="E4:109_1_2_18"/><i><font color="Green" title="E94">A93</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">v</font>
 <b>and </b><br/><a NAME="E5:109_1_2_18"/><i><font color="Green" title="E95">A94</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E6:109_1_2_18"/><i><font color="Green" title="E96">A95</font></i>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2_18"><i><font color="Green" title="E93">A92</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">Vu</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:109_1_2_18"/><i><font color="Green" title="E97">A96</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Vu</font>
 <b>and </b><br/><a NAME="E9:109_1_2_18"/><i><font color="Green" title="E98">A97</font></i>: 
<font color="Maroon" title="c21">Vu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21:109_1_2"><i><font color="Green" title="E81">A39</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_18"><i><font color="Green" title="E95">A94</font></i></a>, <a class="ref" href="rltopsp1.html#T2" target="_self" title="RLTOPSP1:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c23">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E11:109_1_2_18"/><i><font color="Green" title="E99">A98</font></i>: 
( <font color="Maroon" title="c21">Vu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c23">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E23:109_1_2"><i><font color="Green" title="E83">A41</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_18"><i><font color="Green" title="E98">A97</font></i></a></i>;<br/>
<a NAME="E12:109_1_2_18"/><i><font color="Green" title="E100">A99</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E4:109_1_2_18"><i><font color="Green" title="E94">A93</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_18"><i><font color="Green" title="E96">A95</font></i></a>, <a class="txt" href="rltopsp1.html#E8:109_1_2_18"><i><font color="Green" title="E97">A96</font></i></a>, <a class="txt" href="rltopsp1.html#E11:109_1_2_18"><i><font color="Green" title="E99">A98</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self" title="RLTOPSP1:th.3">Th3</a></i>;<br/>
<a NAME="E13:109_1_2_18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2_18"><i><font color="Green" title="E98">A97</font></i></a>, <a class="txt" href="rltopsp1.html#E11:109_1_2_18"><i><font color="Green" title="E99">A98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:109_1_2_18"/>
<font color="Maroon" title="c18">z</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"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2_18"><i><font color="Green" title="E100">A99</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="E42:109_1_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:109_1_2_19"/>
<font color="Maroon" title="c18">z</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"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_19"/><i><font color="Green" title="E93">A100</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Y</font>
 <b>and </b><br/><a NAME="E4:109_1_2_19"/><i><font color="Green" title="E94">A101</font></i>: 
<font color="Maroon" title="c19">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <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="c20">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c21">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E6:109_1_2_19"/><i><font color="Green" title="E95">A102</font></i>: 
<font color="Maroon" title="c19">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font>
 <b>and </b><br/><a NAME="E7:109_1_2_19"/><i><font color="Green" title="E96">A103</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:109_1_2_19"><i><font color="Green" title="E94">A101</font></i></a></i>;<br/>
<a NAME="E8:109_1_2_19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</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">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">u</font>, <font color="Maroon" title="c23">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="E10:109_1_2_19"/><i><font color="Green" title="E97">A104</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">v</font>
 <b>and </b><br/><a NAME="E11:109_1_2_19"/><i><font color="Green" title="E98">A105</font></i>: 
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font>
 <b>and </b><br/><a NAME="E12:109_1_2_19"/><i><font color="Green" title="E99">A106</font></i>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_19"><i><font color="Green" title="E93">A100</font></i></a>, <a class="txt" href="rltopsp1.html#E6:109_1_2_19"><i><font color="Green" title="E95">A102</font></i></a></i>;<br/>
<a NAME="E13:109_1_2_19"/>
<font color="Maroon" title="c21">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E7:109_1_2_19"><i><font color="Green" title="E96">A103</font></i></a></i>;<br/>
<a NAME="E14:109_1_2_19"/><b>then </b>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Vx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2_19"><i><font color="Green" title="E99">A106</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E15:109_1_2_19"/><b>then </b><i><font color="Green" title="E100">A107</font></i>: 
<font color="Maroon" title="c22">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c21">Vx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E11:109_1_2_19"><i><font color="Green" title="E98">A105</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self" title="RLTOPSP1:th.3">Th3</a></i>;<br/>
<a NAME="E16:109_1_2_19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c21">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E7:109_1_2_19"><i><font color="Green" title="E96">A103</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:109_1_2_19"/>
<font color="Maroon" title="c18">z</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"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E10:109_1_2_19"><i><font color="Green" title="E97">A104</font></i></a>, <a class="txt" href="rltopsp1.html#E15:109_1_2_19"><i><font color="Green" title="E100">A107</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="E43:109_1_2"/><b>then </b><i><font color="Green" title="E93">A108</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E41:109_1_2"><i><font color="Green" title="E92">A91</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:109_1_2_20_1"/>
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:109_1_2_20_1"/><b>then </b><i><font color="Green" title="E94">A109</font></i>: 
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">A</font> = <font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>consider </b><font color="Maroon" title="c20">y</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="E5:109_1_2_20_1"/><i><font color="Green" title="E95">A110</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:109_1_2"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:109_1_2_20_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">W</font>
 <b>and </b><br/><a NAME="E8:109_1_2_20_1"/><i><font color="Green" title="E96">A111</font></i>: 
<font color="Maroon" title="c21">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21:109_1_2"><i><font color="Green" title="E81">A39</font></i></a>, <a class="txt" href="rltopsp1.html#E5:109_1_2_20_1"><i><font color="Green" title="E95">A110</font></i></a>, <a class="ref" href="rltopsp1.html#T2" target="_self" title="RLTOPSP1:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c23">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E10:109_1_2_20_1"/><i><font color="Green" title="E97">A112</font></i>: 
( <font color="Maroon" title="c21">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c23">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E23:109_1_2"><i><font color="Green" title="E83">A41</font></i></a>, <a class="txt" href="rltopsp1.html#E8:109_1_2_20_1"><i><font color="Green" title="E96">A111</font></i></a></i>;<br/><a NAME="E11:109_1_2_20_1"/><i><font color="Green" title="E98">A113</font></i>: 
<font color="Maroon" title="c23">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E24:109_1_2"><i><font color="Green" title="E84">A42</font></i></a>, <a class="txt" href="rltopsp1.html#E8:109_1_2_20_1"><i><font color="Green" title="E96">A111</font></i></a>, <a class="txt" href="rltopsp1.html#E10:109_1_2_20_1"><i><font color="Green" title="E97">A112</font></i></a></i>;<br/><a NAME="E12:109_1_2_20_1"/><i><font color="Green" title="E99">A114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c23">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E24:109_1_2"><i><font color="Green" title="E84">A42</font></i></a>, <a class="txt" href="rltopsp1.html#E8:109_1_2_20_1"><i><font color="Green" title="E96">A111</font></i></a>, <a class="txt" href="rltopsp1.html#E10:109_1_2_20_1"><i><font color="Green" title="E97">A112</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c23">Vx</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c25">A</font> = <font color="Maroon" title="c19">A</font>;<br/><b>take </b><font color="Maroon" title="c26">B</font> = <font color="Maroon" title="c24">B</font>;<br/><b>thus </b><a NAME="E14:109_1_2_20_1"/>
<font color="Maroon" title="c25">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E15:109_1_2_20_1"/>
<font color="Maroon" title="c26">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_20_1"><i><font color="Green" title="E96">A111</font></i></a>, <a class="txt" href="rltopsp1.html#E10:109_1_2_20_1"><i><font color="Green" title="E97">A112</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_2_20_1_1"/>
<font color="Maroon" title="c25">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c26">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c27">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_20_1_1"/><i><font color="Green" title="E100">A115</font></i>: 
<font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A</font>
 <b>and </b><br/><a NAME="E4:109_1_2_20_1_1"/><i><font color="Green" title="E101">A116</font></i>: 
<font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">z</font> = <font color="Maroon" title="c27">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_20_1_1"><i><font color="Green" title="E100">A115</font></i></a></i>;<br/><a NAME="E6:109_1_2_20_1_1"/>
<font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</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">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c29">u</font>, <font color="Maroon" title="c30">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="E8:109_1_2_20_1_1"/><i><font color="Green" title="E102">A117</font></i>: 
<font color="Maroon" title="c28">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c30">v</font>
 <b>and </b><br/><a NAME="E9:109_1_2_20_1_1"/><i><font color="Green" title="E103">A118</font></i>: 
<font color="Maroon" title="c29">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E10:109_1_2_20_1_1"/><i><font color="Green" title="E104">A119</font></i>: 
<font color="Maroon" title="c30">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_20_1_1"><i><font color="Green" title="E100">A115</font></i></a></i>;<br/><a NAME="E11:109_1_2_20_1_1"/>
<font color="Maroon" title="c23">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_20_1"><i><font color="Green" title="E96">A111</font></i></a>, <a class="txt" href="rltopsp1.html#E10:109_1_2_20_1"><i><font color="Green" title="E97">A112</font></i></a></i>;<br/><a NAME="E12:109_1_2_20_1_1"/><b>then </b>
<font color="Maroon" title="c30">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E10:109_1_2_20_1_1"><i><font color="Green" title="E104">A119</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E13:109_1_2_20_1_1"/><b>then </b>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c30">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E11:109_1_2_20_1"><i><font color="Green" title="E98">A113</font></i></a>, <a class="ref" href="rltopsp1.html#T26" target="_self" title="RLTOPSP1:th.26">Th26</a></i>;<br/><a NAME="E14:109_1_2_20_1_1"/><b>then </b><i><font color="Green" title="E105">A120</font></i>: 
<font color="Maroon" title="c28">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c30">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="Maroon" title="c22">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c23">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c23">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:109_1_2_20_1_1"><i><font color="Green" title="E101">A116</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self" title="RLTOPSP1:th.3">Th3</a></i>;<br/><font color="Maroon" title="c28">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c30">v</font></span>)</span> = 
<font color="Maroon" title="c29">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c30">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_20_1_1"><i><font color="Green" title="E102">A117</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c29">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c29">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E16:109_1_2_20_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rltopsp1.html#E12:109_1_2_20_1"><i><font color="Green" title="E99">A114</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_20_1_1"><i><font color="Green" title="E103">A118</font></i></a>, <a class="txt" href="rltopsp1.html#E14:109_1_2_20_1_1"><i><font color="Green" title="E105">A120</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:109_1_2_20_1"/>
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c26">B</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:109_1_2_20_1"><i><font color="Green" title="E94">A109</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c19">A</font>, <font color="Maroon" title="c20">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:109_1_2_20"/><i><font color="Green" title="E94">A121</font></i>: 
<font color="Maroon" title="c19">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:109_1_2_20"/><i><font color="Green" title="E95">A122</font></i>: 
<font color="Maroon" title="c20">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E5:109_1_2_20"/><i><font color="Green" title="E96">A123</font></i>: 
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">B</font>
 ;<br/><a NAME="E6:109_1_2_20"/><i><font color="Green" title="E97">A124</font></i>: 
<font color="Maroon" title="c19">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_20"><i><font color="Green" title="E94">A121</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c22">Vx</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> such that </b><br/><a NAME="E8:109_1_2_20"/><i><font color="Green" title="E98">A125</font></i>: 
<font color="Maroon" title="c20">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c22">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c22">Vx</font>
 <b>and </b><br/><a NAME="E9:109_1_2_20"/><i><font color="Green" title="E99">A126</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c22">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:109_1_2_20"><i><font color="Green" title="E95">A122</font></i></a></i>;<br/><a NAME="E10:109_1_2_20"/><i><font color="Green" title="E100">A127</font></i>: 
<font color="Maroon" title="c22">Vx</font> is <a href="rltopsp1.html#V2" title="RLTOPSP1:attr.2">symmetric</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E24:109_1_2"><i><font color="Green" title="E84">A42</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_20"><i><font color="Green" title="E99">A126</font></i></a></i>;<br/><a NAME="E11:109_1_2_20"/><i><font color="Green" title="E101">A128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c22">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c22">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c22">Vx</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E24:109_1_2"><i><font color="Green" title="E84">A42</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_20"><i><font color="Green" title="E99">A126</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_2_20_2"/>
<font color="Maroon" title="c19">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c20">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c23">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_20_2"/><i><font color="Green" title="E102">A129</font></i>: 
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A</font>
 <b>and </b><br/><a NAME="E4:109_1_2_20_2"/><i><font color="Green" title="E103">A130</font></i>: 
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">z</font> = <font color="Maroon" title="c23">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="rltopsp1.html#E3:109_1_2_20"><i><font color="Green" title="E94">A121</font></i></a>, <a class="txt" href="rltopsp1.html#E3:109_1_2_20_2"><i><font color="Green" title="E102">A129</font></i></a></i>;<br/><a NAME="E6:109_1_2_20_2"/>
<font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</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">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u</font>, <font color="Maroon" title="c26">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="E8:109_1_2_20_2"/><i><font color="Green" title="E104">A131</font></i>: 
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">v</font>
 <b>and </b><br/><a NAME="E9:109_1_2_20_2"/><i><font color="Green" title="E105">A132</font></i>: 
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E10:109_1_2_20_2"/><i><font color="Green" title="E106">A133</font></i>: 
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:109_1_2_20"><i><font color="Green" title="E97">A124</font></i></a>, <a class="txt" href="rltopsp1.html#E3:109_1_2_20_2"><i><font color="Green" title="E102">A129</font></i></a></i>;<br/><a NAME="E11:109_1_2_20_2"/>
<font color="Maroon" title="c22">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FVx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E9:109_1_2_20"><i><font color="Green" title="E99">A126</font></i></a></i>;<br/><a NAME="E12:109_1_2_20_2"/><b>then </b>
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E10:109_1_2_20_2"><i><font color="Green" title="E106">A133</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E13:109_1_2_20_2"/><b>then </b>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E10:109_1_2_20"><i><font color="Green" title="E100">A127</font></i></a>, <a class="ref" href="rltopsp1.html#T26" target="_self" title="RLTOPSP1:th.26">Th26</a></i>;<br/><a NAME="E14:109_1_2_20_2"/><b>then </b><i><font color="Green" title="E107">A134</font></i>: 
<font color="Maroon" title="c24">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c26">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="Maroon" title="c21">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c22">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c22">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c22">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_20"><i><font color="Green" title="E98">A125</font></i></a>, <a class="txt" href="rltopsp1.html#E4:109_1_2_20_2"><i><font color="Green" title="E103">A130</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self" title="RLTOPSP1:th.3">Th3</a></i>;<br/><font color="Maroon" title="c24">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c26">v</font></span>)</span> = 
<font color="Maroon" title="c25">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c26">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rltopsp1.html#E8:109_1_2_20_2"><i><font color="Green" title="E104">A131</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c25">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c25">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E16:109_1_2_20_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rltopsp1.html#E11:109_1_2_20"><i><font color="Green" title="E101">A128</font></i></a>, <a class="txt" href="rltopsp1.html#E9:109_1_2_20_2"><i><font color="Green" title="E105">A132</font></i></a>, <a class="txt" href="rltopsp1.html#E14:109_1_2_20_2"><i><font color="Green" title="E107">A134</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E13:109_1_2_20"/><b>then </b>
<font color="Maroon" title="c19">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E14:109_1_2_20"/>
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:109_1_2_20"><i><font color="Green" title="E96">A123</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E45:109_1_2"/><b>then </b>
 <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</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"><font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/><a NAME="E46:109_1_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><a NAME="E47:109_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#T36" title="SETFAM_1:th.36">SETFAM_1:36</a></i>;<br/><a NAME="E48:109_1_2"/><b>then </b>
<font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">Vx</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Olive" title="b2">Vx</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <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> : ( <font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Olive" title="b2">Vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">Vx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E49:109_1_2"/>
<font color="Maroon" title="c2">K</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E43:109_1_2"><i><font color="Green" title="E93">A108</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
