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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font>, <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( ( <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Olive" title="b2">A2</font>,<font color="Olive" title="b3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) iff <font color="Olive" title="b1">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font>,<font color="Olive" title="b3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) iff <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>







<b>thus </b><a NAME="E1:53"/>
( ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) iff <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 <b>by </b><i><a class="ref" href="connsp_1.html#T9" title="CONNSP_1:th.9">CONNSP_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) )</span><br/>

<b>thus </b><a NAME="E2:53_1"/>
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>

<a NAME="E2:53_1_1"/>
( <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:53_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="connsp_1.html#T8" title="CONNSP_1:th.8">CONNSP_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


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