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<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>; <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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/> <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/> <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><b>let </b><font color="Maroon" title="c3">p</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies  <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span><br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">B'</font> = <font color="Maroon" title="c2">B</font> as  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 <font color="Maroon" title="c1">GX</font> ;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">B'</font></span>)</span> 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#T41" title="CONNSP_1:th.41">CONNSP_1:41</a></i>;<br/>
<a NAME="E4:39"/>
 <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2" title="CONNSP_3:func.2">Down</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:39"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="connsp_3.html#T27" target="_self" title="CONNSP_3:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E5:39"/>
 <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:39"><i><font color="Green" title="E23">A2</font></i></a>, <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/>


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