<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">S</font>, <font color="Maroon" title="c7">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> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</font><br/> for <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="c7">T</font>  st  <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="c6">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">T</font> #) &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  holds <br/><font color="Olive" title="b2">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>  st  <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="c6">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">T</font> #) &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c8">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  holds <br/><font color="Olive" title="b1">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="c6">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">T</font> #) &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c8">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  implies <font color="Maroon" title="c9">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <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="c6">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">T</font> #)
 <span class="kw">and </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c8">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="c9">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">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="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q</font> &amp; <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; not <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> implies <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</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="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> or <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">P1</font> = <font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c13">Q1</font> = <font color="Maroon" title="c11">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/><a NAME="E4:9_1"/>
<font color="Maroon" title="c12">P1</font>,<font color="Maroon" title="c13">Q1</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="#E1:9"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="topalg_5.html#T5" target="_self" title="TOPALG_5:th.5">Th5</a></span>;<br/><a NAME="E5:9_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">S</font> or <font color="Maroon" title="c13">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="connsp_1.html#T16" title="CONNSP_1:th.16">CONNSP_1:16</a></span>;<br/><span class="kw">hence </span><a NAME="E6:9_1"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> or <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c7">T</font> )
 ; <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">hence </span><a NAME="E4:9"/>
<font color="Maroon" title="c9">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T16" title="CONNSP_1:th.16">CONNSP_1:16</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>
