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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo6.html#M1" target="_self">SubSpace</a> of <font color="Maroon">FT</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 ;<br/>

<a NAME="E2:18"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:18_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font> implies <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E41</font></i>: 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:18_1"/>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon">X</font> is <a href="fin_topo.html#V2">symmetric</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/>


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