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<b>let </b><font color="Maroon" title="c1">T</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">A</font>, <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="c1">T</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span><br/>





<b>thus </b><a NAME="E1:11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  )</span><br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span><br/>

<a NAME="E2:11_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">px</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E1:11_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E4:11_1"/>
(  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="#E2:11_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="fin_topo.html#T12" title="FIN_TOPO:th.12">FIN_TOPO:12</a></i>;<br/>
<a NAME="E5:11_1"/><b>then </b>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T12" title="FIN_TOPO:th.12">FIN_TOPO:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">B</font></span>)</span> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">px</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:11"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E2:11"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="fin_topo.html#T12" title="FIN_TOPO:th.12">FIN_TOPO:12</a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E6:11"/>
 <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E4:11"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a>  )
 
<b>by </b><i><a class="txt" href="#E5:11"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="fin_topo.html#T12" title="FIN_TOPO:th.12">FIN_TOPO:12</a></i>;<br/>
<b>hence </b><a NAME="E8:11"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</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="c3">B</font> <a href="fin_topo.html#K8" title="FIN_TOPO:func.8">^i</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
