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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">r</font> holds <br/><font color="Maroon" title="c1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">r</font> implies <font color="Maroon" title="c1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/>

<a NAME="E2:17"/>
<font color="Maroon" title="c1">A</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">w1</font>, <font color="Maroon" title="c5">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  not <font color="Maroon" title="c5">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Maroon" title="c5">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c5">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Maroon" title="c5">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">thus </span><a NAME="E3:17_1"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Maroon" title="c5">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="topreal3.html#T28" title="TOPREAL3:th.28">TOPREAL3:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:17"/>
<font color="Maroon" title="c1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1.html#T9" title="JORDAN1:th.9">JORDAN1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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