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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Maroon" title="c1">GX</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Olive" title="b2">C</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">C</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">A</font>, <font color="Maroon" title="c3">C</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">GX</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>  &amp; <font color="Maroon" title="c2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c3">C</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> implies <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</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="E1:44"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">GX</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 <span class="kw">and </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">C1</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:44"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E6:44"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c4">C1</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="connsp_1.html#D6" target="_self" title="CONNSP_1:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">C2</font> = <font color="Maroon" title="c4">C1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>;<br/>
<a NAME="E8:44"/>
<font color="Maroon" title="c4">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E9:44"/><span class="kw">then </span>
<font color="Maroon" title="c4">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E10:44"/><span class="kw">then </span>
<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="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T12" title="SUBSET_1:th.12">SUBSET_1:12</a></span>;<br/>
<a NAME="E11:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T3" title="PRE_TOPC:th.3">PRE_TOPC:3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:44_1"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:44"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><a NAME="E2:44_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E3:44_1"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c3">C</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="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="connsp_1.html#T23" target="_self" title="CONNSP_1:th.23">Th24</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp;  not <font color="Maroon" title="c6">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="c1">GX</font> implies <font color="Maroon" title="c7">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="c1">GX</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:44_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:44_1"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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="c1">GX</font> or <font color="Maroon" title="c7">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="c1">GX</font> )</span><br/><a NAME="E6:44_1"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E7:44_1"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="connsp_1.html#T1" target="_self" title="CONNSP_1:th.1">Th2</a></span>;<br/><div><span class="lab"><font color="Green" title="E44">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:44_1_1"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c7">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><span class="kw">assume </span><a NAME="E2:44_1_1"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="c1">GX</font></span><br/><a NAME="E3:44_1_1"/>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_1"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><a NAME="E4:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1_1"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E5:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E5:44_1_1"><span class="lab"><font color="Green" title="E47">A16</font></span></a></span>
;<br/><a NAME="E7:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E8:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><a NAME="E9:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">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> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">C1P1</font> = <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> ;<br/><a NAME="E11:44_1_1"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E12:44_1_1"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">P</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="#E1:44"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="connsp_1.html#T20" target="_self" title="CONNSP_1:th.20">Th21</a></span>;<br/><a NAME="E13:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c8">C1P1</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#T23" target="_self" title="CONNSP_1:th.23">Th24</a></span>;<br/><a NAME="E14:44_1_1"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E15:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">C1P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E13:44_1_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="connsp_1.html#D5" target="_self" title="CONNSP_1:def.5">Def5</a></span>;<br/><a NAME="E16:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E17:44_1_1"/>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E18:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</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="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:44_1_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E19:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:44_1_1"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E20:44_1_1"/>
<font color="Maroon" title="c6">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="c1">GX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:44_1"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_1_2"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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="c1">GX</font></span><br/><a NAME="E2:44_1_2"/>
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_1"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><a NAME="E3:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E4:44_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E4:44_1_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a></span>
;<br/><a NAME="E6:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E7:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><a NAME="E8:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">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> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">C1Q1</font> = <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> ;<br/><a NAME="E10:44_1_2"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Q</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="#E1:44"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="connsp_1.html#T20" target="_self" title="CONNSP_1:th.20">Th21</a></span>;<br/><a NAME="E11:44_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c8">C1Q1</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#T23" target="_self" title="CONNSP_1:th.23">Th24</a></span>;<br/><a NAME="E12:44_1_2"/>
<font color="Maroon" title="c4">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E13:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">C1Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E11:44_1_2"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="connsp_1.html#D5" target="_self" title="CONNSP_1:def.5">Def5</a></span>;<br/><a NAME="E14:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E15:44_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</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="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44_1"><span class="lab"><font color="Green" title="E45">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="E16:44_1_2"/>
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E17:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:44_1_2"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E18:44_1_2"/>
<font color="Maroon" title="c7">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="c1">GX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:44_1"/>
( <font color="Maroon" title="c6">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="c1">GX</font> or <font color="Maroon" title="c7">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="c1">GX</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E11:44"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E8:44_1"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="connsp_1.html#T16" target="_self" title="CONNSP_1:th.16">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:44"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</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#T15" target="_self" title="CONNSP_1:th.15">Th16</a></span>; <a class="txt" 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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