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

<b>let </b><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span><br/> for <font color="Olive" title="b2">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/>( <font color="Olive" title="b1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Olive" title="b2">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span>)</span>; <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">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c3">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Olive" title="b1">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> )</span><br/><b>let </b><font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> implies ( <font color="Maroon" title="c3">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> ) )</span><br/>







<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</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> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c3">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> implies <font color="Maroon" title="c3">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon" title="c3">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a></span><br/><a NAME="E2:62_1"/><b>then </b>
<font color="Maroon" title="c5">Z</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/><b>hence </b><a NAME="E3:62_1"/>
<font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="rcomp_3.html#T52" target="_self" title="RCOMP_3:th.52">Th52</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>

<b>assume </b><a NAME="E4:62"/>
<font color="Maroon" title="c4">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a>
 ; <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> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/>

<a NAME="E5:62"/><b>then </b>
<font color="Maroon" title="c5">Z</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="rcomp_3.html#T52" target="_self" title="RCOMP_3:th.52">Th52</a></i>;<br/>
<b>hence </b><a NAME="E6:62"/>
<font color="Maroon" title="c3">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</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>
