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

<b>let </b><font color="Maroon" title="c1">GX</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>  ex <font color="Olive" title="b3">GY</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> st <br/>( <font color="Olive" title="b3">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> &amp;  ex <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b3">GY</font>,<font color="Maroon" title="c1">GX</font> st <br/>( <font color="Olive" title="b4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">x</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="Olive" title="b4">f</font> &amp; <font color="Olive" title="b2">y</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="Olive" title="b4">f</font> ) )
 ;<br/>

<a NAME="E2:1"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font>
 <b>and </b><br/><a NAME="E3:1_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font>
 <b>and </b><br/><a NAME="E4:1_1"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:1_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="txt" href="jordan1.html#E1:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="jordan1.html#E2:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">B</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="txt" href="jordan1.html#E1:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="jordan1.html#E3:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">GY</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><b> such that </b><br/><a NAME="E11:1_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c8">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>and </b><br/><a NAME="E12:1_1"/><i><font color="Green" title="E9">A8</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">GY</font>,<font color="Maroon" title="c1">GX</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c7">y</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="Olive" title="b1">f</font> )
 <b>by </b><i><a class="txt" href="jordan1.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">GY</font>,<font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E14:1_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c9">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E15:1_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c5">x</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="c9">f</font>
 <b>and </b><br/><a NAME="E16:1_1"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c7">y</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="c9">f</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E12:1_1"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<a NAME="E17:1_1"/><i><font color="Green" title="E13">A0</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</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/>
<a NAME="E18:1_1"/><b>then </b><i><font color="Green" title="E14">A12</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan1.html#E4:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="jordan1.html#E14:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E19:1_1"/><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan1.html#E17:1_1"><i><font color="Green" title="E13">A0</font></i></a>, <a class="txt" href="jordan1.html#E5:1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="jordan1.html#E14:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E20:1_1"/>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c8">GY</font>
 
<b>by </b><i><a class="ref" href="tops_2.html#T52" title="TOPS_2:th.52">TOPS_2:52</a></i>;<br/>
<a NAME="E21:1_1"/><b>then </b><i><font color="Green" title="E16">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c8">GY</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="relat_1.html#T175" title="RELAT_1:th.175">RELAT_1:175</a></i>;<br/>
<a NAME="E22:1_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</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="txt" href="jordan1.html#E2:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="jordan1.html#E15:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E24:1_1"/><b>then </b><i><font color="Green" title="E17">A15</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c8">GY</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T173" title="RELAT_1:th.173">RELAT_1:173</a></i>;<br/>
<a NAME="E25:1_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</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="txt" href="jordan1.html#E3:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="jordan1.html#E16:1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E26:1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E27:1_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c8">GY</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T173" title="RELAT_1:th.173">RELAT_1:173</a></i>;<br/>
<a NAME="E28:1_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="jordan1.html#E11:1_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="jordan1.html#E18:1_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="jordan1.html#E19:1_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="jordan1.html#E21:1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="jordan1.html#E24:1_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="connsp_1.html#T12" title="CONNSP_1:th.12">CONNSP_1:12</a></i>;<br/>
<a NAME="E29:1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">B</font></span>)</span> <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="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E30:1_1"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <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="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>;<br/>
<a NAME="E31:1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T173" title="RELAT_1:th.173">RELAT_1:173</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">u1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E33:1_1"/><i><font color="Green" title="E18">A16</font></i>: 
( <font color="Maroon" title="c10">u1</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="c9">f</font> &amp; <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">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>thus </b><a NAME="E34:1_1"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E33:1_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T12" title="CONNSP_1:th.12">CONNSP_1:12</a></i>;<br/>


</div>
