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<span class="kw">let </span><font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/>for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="b3">q</font> holds <br/><font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> ( <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="b2">q</font> holds <br/><font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="b2">q</font> holds <br/><font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">F</font> =  <a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</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"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c5">q</font> implies <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c5">q</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="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E4:17"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#T8" target="_self" title="T_0TOPSP:th.8">Th8</a></span>;<br/>
<a NAME="E5:17"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</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="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="borsuk_1.html#T70" title="BORSUK_1:th.70">BORSUK_1:70</a></span>;<br/>
<a NAME="E6:17"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">p</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="t_0topsp.html#D3" target="_self" title="T_0TOPSP:def.3">Def3</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/>


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