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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X'</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fintopo6.html#M1" title="FINTOPO6:mode.1">SubSpace</a> of <font color="Maroon" title="c1">FT</font><br/> for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">Q1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font><br/> for <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X'</font>  st <font color="Olive" title="b4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P1</font> &amp; <font color="Olive" title="b5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Q1</font> &amp; <font color="Olive" title="b4">P</font>,<font color="Olive" title="b5">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  holds <br/><font color="Olive" title="b2">P1</font>,<font color="Olive" title="b3">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> </span><br/><b>let </b><font color="Maroon" title="c2">X'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fintopo6.html#M1" title="FINTOPO6:mode.1">SubSpace</a> of <font color="Maroon" title="c1">FT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">FT</font><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X'</font>  st <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> &amp; <font color="Olive" title="b4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q1</font> &amp; <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  holds <br/><font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> </span><br/><b>let </b><font color="Maroon" title="c3">P1</font>, <font color="Maroon" title="c4">Q1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">X'</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font> &amp; <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  holds <br/><font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> </span><br/><b>let </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font> &amp; <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font> &amp; <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  implies <font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  or <font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  )</span><br/>

<b>assume </b><a NAME="E3:23"/>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> </span><br/>

<a NAME="E4:23"/><b>then </b>
( <font color="Maroon" title="c5">P</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c5">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  )
 
<b>by </b><i><a class="ref" href="fintopo4.html#D1" title="FINTOPO4:def.1">FINTOPO4:def 1</a></i>;<br/>
<a NAME="E5:23"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">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">Q</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">P2</font> = <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c8">Q2</font> = <font color="Maroon" title="c6">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="ref" href="fintopo6.html#T10" target="_self" title="FINTOPO6:th.10">Th10</a></i>;<br/>
<i><font color="Green" title="E18">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <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">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">Q</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T13" target="_self" title="FINTOPO6:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</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">X'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Q2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<font color="Maroon" title="c5">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">Q</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> = 
<font color="Maroon" title="c5">P</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">X'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">Q2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fintopo6.html#T13" target="_self" title="FINTOPO6:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</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">X'</font></span>)</span></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="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<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"><font color="Maroon" title="c8">Q2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E9:23"/><b>then </b>
( <font color="Maroon" title="c7">P2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Q2</font> &amp; <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> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  )
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:23"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="fintopo6.html#E7:23"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
<font color="Maroon" title="c3">P1</font>,<font color="Maroon" title="c4">Q1</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:23"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="fintopo6.html#E2:23"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="fintopo4.html#D1" title="FINTOPO4:def.1">FINTOPO4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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