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<b>let </b><font color="Maroon" title="c6">S</font>, <font color="Maroon" title="c7">T</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><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">T</font> #)
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c6">S</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D4" title="CONNSP_2:def.4">CONNSP_2:def 4</a><br/><b>let </b><font color="Maroon" title="c9">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D3" title="CONNSP_2:def.3">CONNSP_2:def 3</a><br/>



<b>reconsider </b><font color="Maroon" title="c10">U1</font> = <font color="Maroon" title="c9">U</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c8">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c6">S</font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:14"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="connsp_2.html#D4" title="CONNSP_2:def.4">CONNSP_2:def 4</a></i>;<br/>
<b>assume </b><a NAME="E6:14"/>
<font color="Maroon" title="c9">U</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">t</font>
 ;<br/>

<a NAME="E7:14"/><b>then </b>
<font color="Maroon" title="c10">U1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="topalg_5.html#T8" target="_self" title="TOPALG_5:th.8">Th8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</font><b> such that </b><br/><a NAME="E9:14"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c12">V1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c11">s</font>
 <b>and </b><br/><a NAME="E10:14"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c12">V1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E11:14"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c12">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">U1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:14"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="connsp_2.html#D3" title="CONNSP_2:def.3">CONNSP_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">V</font> = <font color="Maroon" title="c12">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">V</font>
;<br/>

<b>thus </b><a NAME="E13:14"/>
<font color="Maroon" title="c13">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="topalg_5.html#E9:14"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="topalg_5.html#T8" target="_self" title="TOPALG_5:th.8">Th8</a></i>;<br/>

<b>thus </b><a NAME="E14:14"/>
<font color="Maroon" title="c13">V</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:14"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="topalg_5.html#E10:14"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self" title="TOPALG_5:th.7">Th7</a></i>;<br/>

<b>thus </b><a NAME="E15:14"/>
<font color="Maroon" title="c13">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:14"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>


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