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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="topgrp_1.html#NM4" title="TOPGRP_1:NM.4">TopGroup</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>   ex <font color="Olive" title="b3">A</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> st <font color="Olive" title="b3">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font> ) implies <font color="Maroon" title="c1">T</font> is  <a href="topgrp_1.html#V3" title="TOPGRP_1:attr.3">UnContinuous</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:76"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>   ex <font color="Olive" title="b3">A</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> st <font color="Olive" title="b3">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</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="c1">T</font> is  <a href="topgrp_1.html#V3" title="TOPGRP_1:attr.3">UnContinuous</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E2:76"/>
for <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">W</font>  ex <font color="Olive" title="b3">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">W</font> st <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <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">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>  ex <font color="Olive" title="b2">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">a</font> st <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">a</font> st <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font></span><br/>



<a NAME="E1:76_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" title="GROUP_1:def.7">GROUP_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">a</font><span class="kw"> such that </span><br/><a NAME="E3:76_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font></span><br/>

<span class="kw">thus </span><a NAME="E4:76_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="topgrp_1.html#T10" target="_self" title="TOPGRP_1:th.10">Th10</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:76"/>
 <a href="group_1.html#K3" title="GROUP_1:func.3">inverse_op</a> <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="topgrp_1.html#D7" target="_self" title="TOPGRP_1:def.7">TOPGRP_1:def 7</a> <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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