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

<b>let </b><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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">GX</font> st <br/>for <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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a>  iff 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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a>  )</span><br/>

<div><i><font color="Green" title="E29">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c2">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><a NAME="E2:34_1"/><i><font color="Green" title="E30">A2</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">GX</font> holds  <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> </span><br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> </span><br/><a NAME="E3:34_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 <b>by </b><i><a class="txt" href="#E2:34_1"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">C1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">C1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E6:34_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C1</font>
 <b>and </b><br/><a NAME="E7:34_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C1</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#D4" target="_self" title="CONNSP_1:def.4">Def4</a></i>;<br/><a NAME="E8:34_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 <b>by </b><i><a class="txt" href="#E2:34_1"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">C2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E10:34_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c6">C2</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E11:34_1"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C2</font>
 <b>and </b><br/><a NAME="E12:34_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C2</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#D4" target="_self" title="CONNSP_1:def.4">Def4</a></i>;<br/><a NAME="E13:34_1"/>
<font color="Maroon" title="c5">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">C2</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>by </b><i><a class="txt" href="#E6:34_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E11:34_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E14:34_1"/><b>then </b>
<font color="Maroon" title="c5">C1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">C2</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="E15:34_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c5">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C2</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="#E5:34_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E10:34_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self" title="CONNSP_1:th.2">Th2</a>, <a class="ref" href="connsp_1.html#T18" target="_self" title="CONNSP_1:th.18">Th18</a></i>;<br/><a NAME="E16:34_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C2</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C2</font> )
 <b>by </b><i><a class="txt" href="#E7:34_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E12:34_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E17:34_1"/>
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 <b>by </b><i><a class="txt" href="#E15:34_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="connsp_1.html#D4" target="_self" title="CONNSP_1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2"/><i><font color="Green" title="E30">A10</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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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">GX</font> st <br/>for <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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> </span><br/><b>consider </b><font color="Maroon" title="c2">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>;<br/><a NAME="E3:34_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font> holds  <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 <b>by </b><i><a class="txt" href="#E1:34_2"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:34_2"/>
 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">GX</font> st <br/>for <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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
(  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">GX</font> st <br/>for <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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a>  iff 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> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="connsp_1.html#R2" title="CONNSP_1:pred.2">are_joined</a>  )
 <b>by </b><i><a class="txt" href="#E1:34"><i><font color="Green" title="E29">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
