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<b>let </b><font color="Maroon" title="c1">X</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="c1">X</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> implies for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">S</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">X</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"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">S</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E3:22"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">S</font> implies <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E21">A4</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">S</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>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:22_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="connsp_2.html#D4" target="_self" title="CONNSP_2:def.4">Def4</a></i>;<br/><a NAME="E4:22_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="connsp_2.html#T13" target="_self" title="CONNSP_2:th.13">Th13</a></i>;<br/><a NAME="E5:22_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T5" target="_self" title="CONNSP_2:th.5">Th5</a>, <a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></i>;<br/><a NAME="E6:22_1"/><b>then </b>
<font color="Maroon" title="c2">S</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:22_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E3:22_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E4:22_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="connsp_2.html#T19" target="_self" title="CONNSP_2:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E7:22_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1" target="_self" title="CONNSP_2:def.1">Def1</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>
<a NAME="E5:22"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:22"/>
<font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</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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