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<span class="kw">let </span><font color="Maroon" title="c1">TM</font> be   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</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">Am</font>, <font color="Olive" title="b2">Bm</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b1">Am</font>,<font color="Olive" title="b2">Bm</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  holds <br/> ex <font color="Olive" title="b3">L</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <font color="Olive" title="b3">L</font> <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Olive" title="b1">Am</font>,<font color="Olive" title="b2">Bm</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Am</font>, <font color="Maroon" title="c3">Bm</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</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">Am</font>,<font color="Maroon" title="c3">Bm</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies  ex <font color="Olive" title="b1">L</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <font color="Olive" title="b1">L</font> <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Maroon" title="c2">Am</font>,<font color="Maroon" title="c3">Bm</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:55"/>
<font color="Maroon" title="c2">Am</font>,<font color="Maroon" title="c3">Bm</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</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">  ex <font color="Olive" title="b1">L</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font> st <font color="Olive" title="b1">L</font> <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Maroon" title="c2">Am</font>,<font color="Maroon" title="c3">Bm</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">U</font>, <font color="Maroon" title="c5">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><br/><a NAME="E3:55"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
( <font color="Maroon" title="c2">Am</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">U</font> &amp; <font color="Maroon" title="c3">Bm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c4">U</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="metrizts.html#E40"><span class="lab"><font color="Green" title="E32">Lm13</font></span></a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Maroon" title="c2">Am</font>,<font color="Maroon" title="c3">Bm</font></span><br/>

<span class="kw">thus </span><a NAME="E4:55"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">W</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="metrizts.html#R2" title="METRIZTS:pred.2">separates</a> <font color="Maroon" title="c2">Am</font>,<font color="Maroon" title="c3">Bm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E33">A1</font></span></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"> verum</span><br/>


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