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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:18"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:18"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R3" title="CONNSP_1:pred.3">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R3" title="CONNSP_1:pred.3">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/><span class="kw">assume </span><a NAME="E1:18_1"/>
( <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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c9">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <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="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="topalg_5.html#T10" target="_self" title="TOPALG_5:th.10">Th10</a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</a></span>;<br/><a NAME="E3:18_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E4:18_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T139" title="RELAT_1:th.139">RELAT_1:139</a></span>;<br/><a NAME="E5:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">B</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="txt" href="#E3:18"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="connsp_2.html#T18" title="CONNSP_2:th.18">CONNSP_2:18</a></span>;<br/><span class="kw">hence </span><a NAME="E6:18_1"/>
<font color="Maroon" title="c10">B</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="txt" href="#E2:18"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T18" title="CONNSP_2:th.18">CONNSP_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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