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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> iff  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a> )</span><br/>

<a NAME="E1:15"/>
<font color="Maroon" title="c6">T</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">T</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E21">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">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">T</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font></span>)</span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a> implies <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:15_1"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a></span><br/><a NAME="E2:15_1"/><b>then </b>
<font color="Maroon" title="c6">T</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> <font color="Maroon" title="c6">T</font></span>)</span> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="topalg_5.html#T12" target="_self" title="TOPALG_5:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:15_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 <b>by </b><i><a class="ref" href="connsp_2.html#D6" title="CONNSP_2:def.6">CONNSP_2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:15"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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></span><br/>

<a NAME="E5:15"/><b>then </b>
<font color="Maroon" title="c6">T</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> <font color="Maroon" title="c6">T</font></span>)</span> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6" title="CONNSP_2:def.6">CONNSP_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon" title="c6">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="topalg_5.html#T12" target="_self" title="TOPALG_5:th.12">Th12</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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