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<span class="kw">let </span><font color="Maroon" title="c6">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>; <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">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">T</font>  st <font color="Maroon" title="c6">T</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>  holds <br/> <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="Olive" title="b1">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">S</font> #) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">T</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="c6">T</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>  implies  <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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>  )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">S</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="c6">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<a NAME="E2:16"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c8">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D1" title="TSEP_1:def.1">TSEP_1:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16"/>
<font color="Maroon" title="c6">T</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> </span><br/>

<a NAME="E4:16"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">S</font> is  <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="topalg_5.html#T11" target="_self" title="TOPALG_5:th.11">Th11</a>, <a class="ref" href="connsp_2.html#T23" title="CONNSP_2:th.23">CONNSP_2:23</a></span>;<br/>
<a NAME="E5:16"/><span class="kw">then </span>
<font color="Maroon" title="c7">S</font> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T13" target="_self" title="TOPALG_5:th.13">Th13</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span>
 <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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T12" target="_self" title="TOPALG_5:th.12">Th12</a></span>;<br/>
<a NAME="E7:16"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is  <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T13" target="_self" title="TOPALG_5:th.13">Th13</a></span>;<br/>
<a NAME="E8:16"/><span class="kw">then </span>
<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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #)</span>)</span> is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D6" title="CONNSP_2:def.6">CONNSP_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16"/>
 <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">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> #) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</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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