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<span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <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>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T</font><br/> for <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T</font> #)  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">D</font> holds <br/>( <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  iff <font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">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">C</font>, <font color="Olive" title="b2">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T</font> #)  st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">D</font> holds <br/>( <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  iff <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">C</font>, <font color="Maroon" title="c8">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">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="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> implies ( <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  iff <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</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">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  iff <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>

<a NAME="E2:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A2</font></span>: 
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">C</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T80" title="TOPS_3:th.80">TOPS_3:80</a></span>;<br/>
<div><span class="kw">hereby </span> <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="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:6_1"/>
<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ; <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="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><a NAME="E2:6_1"/><span class="kw">then </span>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_1"/>
<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</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">assume </span><a NAME="E4:6"/>
<font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">D</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ; <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">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/>

<a NAME="E5:6"/><span class="kw">then </span>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">D</font> &amp; <font color="Maroon" title="c7">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6"/>
<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_1:def 1</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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