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<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>] )
 <b>and </b><br/><a NAME="E2:51_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>] )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:51_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:51_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:51_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E5:51_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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