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

<span class="kw">let </span><font color="Maroon" title="c1">GX</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">X9</font> being   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">GX</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><br/> for <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X9</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> holds <br/>( <font color="Olive" title="b2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <font color="Olive" title="b3">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="c2">X9</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">GX</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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</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="c2">X9</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> holds <br/>( <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <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="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</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="c2">X9</font>  st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/>( <font color="Maroon" title="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <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="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</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="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> implies ( <font color="Maroon" title="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <font color="Maroon" title="c4">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</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="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <font color="Maroon" title="c4">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">GX9</font> = <font color="Maroon" title="c1">GX</font>, <font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c2">X9</font> as   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> ;<br/>
<a NAME="E3:27"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">B9</font> = <font color="Maroon" title="c4">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">A9</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">GX9</font> ;<br/>
<a NAME="E6:27"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<a NAME="E7:27"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
;<br/>
<div><span class="lab"><font color="Green" title="E26">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1"/>
not <font color="Maroon" title="c3">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"> not <font color="Maroon" title="c4">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><a NAME="E2:27_1"/><span class="kw">then </span>
not <font color="Maroon" title="c5">GX9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A9</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D3" target="_self" title="CONNSP_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">P</font>, <font color="Maroon" title="c10">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:27_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:27_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c9">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E8:27_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c10">Q</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E9:27_1"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">P1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><span class="kw"> such that </span><br/><a NAME="E11:27_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c11">P1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E12:27_1"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">P11</font> = <font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</font> ;<br/><a NAME="E14:27_1"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c12">P11</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">PP</font> = <font color="Maroon" title="c9">P</font>, <font color="Maroon" title="c14">QQ</font> = <font color="Maroon" title="c10">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><a NAME="E16:27_1"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E17:27_1"/>
<font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E18:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E12:27_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E16:27_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E19:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E20:27_1"/>
<font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E21:27_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E12:27_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E22:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">P11</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">PP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E19:27_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E23:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c13">PP</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">Q1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><span class="kw"> such that </span><br/><a NAME="E25:27_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon" title="c15">Q1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E26:27_1"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
<font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">Q11</font> = <font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</font> ;<br/><a NAME="E28:27_1"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E29:27_1"/>
<font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E30:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E26:27_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E28:27_1"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E31:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E32:27_1"/>
<font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E33:27_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E26:27_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E34:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:27_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E35:27_1"/>
<font color="Maroon" title="c16">Q11</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:27_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><a NAME="E36:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">QQ</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E34:27_1"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><a NAME="E37:27_1"/><span class="kw">then </span>
not <font color="Maroon" title="c6">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">B9</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E9:27_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E23:27_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E38:27_1"/>
not <font color="Maroon" title="c4">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#D3" target="_self" title="CONNSP_1:def.3">Def3</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_2"/>
not <font color="Maroon" title="c4">B</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"> not <font color="Maroon" title="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><a NAME="E2:27_2"/><span class="kw">then </span>
not <font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D3" target="_self" title="CONNSP_1:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">P</font>, <font color="Maroon" title="c10">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:27_2"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:27_2"/><span class="lab"><font color="Green" title="E28">A24</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:27_2"/><span class="lab"><font color="Green" title="E29">A25</font></span>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:27_2"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
<font color="Maroon" title="c9">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E8:27_2"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<font color="Maroon" title="c10">Q</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E9:27_2"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">QQ</font> = <font color="Maroon" title="c10">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">PP</font> = <font color="Maroon" title="c9">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">P1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</font><span class="kw"> such that </span><br/><a NAME="E13:27_2"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon" title="c13">P1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E14:27_2"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
<font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_2"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">Q1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</font><span class="kw"> such that </span><br/><a NAME="E16:27_2"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<font color="Maroon" title="c14">Q1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E17:27_2"/><span class="lab"><font color="Green" title="E36">A32</font></span>: 
<font color="Maroon" title="c14">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">Q2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><span class="kw"> such that </span><br/><a NAME="E19:27_2"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c15">Q2</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E20:27_2"/><span class="lab"><font color="Green" title="E38">A34</font></span>: 
<font color="Maroon" title="c15">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:27_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><font color="Maroon" title="c15">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c15">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c11">QQ</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E17:27_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="txt" href="#E20:27_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
;<br/><a NAME="E22:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A35</font></span>: 
<font color="Maroon" title="c11">QQ</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:27_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">P2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><span class="kw"> such that </span><br/><a NAME="E24:27_2"/><span class="lab"><font color="Green" title="E40">A36</font></span>: 
<font color="Maroon" title="c16">P2</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E25:27_2"/><span class="lab"><font color="Green" title="E41">A37</font></span>: 
<font color="Maroon" title="c16">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27_2"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><font color="Maroon" title="c16">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c16">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X9</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c12">PP</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E14:27_2"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="txt" href="#E25:27_2"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
;<br/><a NAME="E27:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">PP</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:27_2"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></span>;<br/><a NAME="E28:27_2"/><span class="kw">then </span>
not <font color="Maroon" title="c5">GX9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">A9</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E4:27_2"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="txt" href="#E5:27_2"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E6:27_2"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E9:27_2"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E22:27_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E29:27_2"/>
not <font color="Maroon" title="c3">A</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#D3" target="_self" title="CONNSP_1:def.3">Def3</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">hence </span><a NAME="E10:27"/>
( <font color="Maroon" title="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  iff <font color="Maroon" title="c4">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="txt" href="#E8:27"><span class="lab"><font color="Green" title="E26">A5</font></span></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>
