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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )</span><br/>



<a NAME="E1:9"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">V</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K2" title="T_0TOPSP:func.2">Indiscernible</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D7" onmouseover="rs('borsuk_1/D7')" onmouseout="rh()">BORSUK_1:def 7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:9"/>
( <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T27" onmouseover="rs('borsuk_1/T27')" onmouseout="rh()">BORSUK_1:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:9"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E4:9"/><span class="kw">then </span>
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K2" title="T_0TOPSP:func.2">Indiscernible</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T27" onmouseover="rs('borsuk_1/T27')" onmouseout="rh()">BORSUK_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9"/>
<font color="Maroon" title="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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