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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">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">P</font>, <font color="Olive" title="b3">Q</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="b4">P1</font>, <font color="Olive" title="b5">Q1</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P1</font> &amp; <font color="Olive" title="b3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">Q1</font> &amp; <font color="Olive" title="b2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">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="Olive" title="b1">X9</font> &amp; <font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  holds <br/><font color="Olive" title="b4">P1</font>,<font color="Olive" title="b5">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</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">P1</font>, <font color="Olive" title="b4">Q1</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P1</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q1</font> &amp; <font color="Olive" title="b1">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">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> &amp; <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  holds <br/><font color="Olive" title="b3">P1</font>,<font color="Olive" title="b4">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">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">  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">Q1</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  holds <br/><font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">P1</font>, <font color="Maroon" title="c6">Q1</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  implies <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font>
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q1</font>
 <span class="kw">and </span><br/><a NAME="E3:7"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  or <font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>

<a NAME="E4:7"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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="E5:7"/>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">P2</font> = <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c8">Q2</font> = <font color="Maroon" title="c4">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:7"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Q</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">and </span><br/><a NAME="E8:7"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>,<a class="ref" href="connsp_1.html#D1" target="_self" title="CONNSP_1:def.1">CONNSP_1:def 1</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="c5">P1</font>,<font color="Maroon" title="c6">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/>

<font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">Q2</font></span>)</span> = 
<font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T17" title="PRE_TOPC:th.17">PRE_TOPC:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P2</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> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E10:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c7">P2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E11:7"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font></span>)</span> <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="ref" href="pre_topc.html#T17" title="PRE_TOPC:th.17">PRE_TOPC:17</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q2</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Q2</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E13:7"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">P2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E12">A5</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="E14:7"/>
<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">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="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E10:7"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="connsp_1.html#D1" target="_self" title="CONNSP_1:def.1">Def1</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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