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<b>let </b><font color="Maroon">Z1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="fintopo6.html#M1" target="_self">SubSpace</a> of <font color="Maroon">T</font>, <font color="Maroon">Z2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="fintopo6.html#M1" target="_self">SubSpace</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:15_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">R1</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Z1</font>, <font color="Maroon">R2</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Z2</font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">P</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:15_1_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">R1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z2</font> <b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">R1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> = 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z1</font>

<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z1</font>
<b>by </b><i/>
<br/>.= 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">z2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">R2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_1_2"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z2</font>
 <b>by </b><i>, , <a class="ref" href="relset_1.html#T54">RELSET_1:54</a></i>;<br/>


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