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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">T</font> is <a href="connsp_2.html#V1">locally_connected</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:17"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D1">TSEP_1:def 1</a></i>;<br/>
<a NAME="E4:17"/><b>then </b>
<font color="Maroon">A</font> is <a href="connsp_2.html#V2">locally_connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#T23">CONNSP_2:23</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> is <a href="connsp_2.html#V2">locally_connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:17"/><b>then </b>
<font color="Maroon">S</font> is <a href="connsp_2.html#V1">locally_connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:17"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) is <a href="connsp_2.html#V1">locally_connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:17"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) is <a href="connsp_2.html#V2">locally_connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:17"/><b>then </b>
<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)</span>)</span> is <a href="connsp_2.html#V1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6">CONNSP_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E10:17"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) is <a href="connsp_2.html#V1">locally_connected</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>


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