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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c7">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  &amp; <font color="Maroon" title="c6">S</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> implies <font color="Maroon" title="c7">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> )</span><br/>

<b>given </b><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c7">T</font><b> such that </b><a NAME="E2:18"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">S</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> or <font color="Maroon" title="c7">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> )</span><br/>

<b>assume </b><a NAME="E3:18"/><i><font color="Green" title="E24">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">T</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">A</font> be  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="c7">T</font>; <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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>  st <font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c9">A</font> holds <br/><font color="Olive" title="b1">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><b>let </b><font color="Maroon" title="c10">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c9">A</font> implies <font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:18_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:18_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c9">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><a NAME="E3:18_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:18"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E4:18_1"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
not <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T174" title="RELAT_1:th.174">RELAT_1:174</a></i>;<br/><a NAME="E5:18_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:18"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="topalg_5.html#E1:18_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</a></i>;<br/><a NAME="E6:18_1"/>
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:18"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="topalg_5.html#E2:18_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="topalg_5.html#T10" target="_self" title="TOPALG_5:th.10">Th10</a></i>;<br/><a NAME="E7:18_1"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:18"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="topalg_5.html#E4:18_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="topalg_5.html#E5:18_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="connsp_2.html#T24" title="CONNSP_2:th.24">CONNSP_2:24</a></i>;<br/><b>hence </b><a NAME="E8:18_1"/>
<font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:18"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</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>hence </b><a NAME="E5:18"/>
<font color="Maroon" title="c7">T</font> 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#T24" title="CONNSP_2:th.24">CONNSP_2:24</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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