<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">R</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> 2</span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">R</font> <a href="topreal4.html#NR3" target="_self">is_Region</a> 
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<b>set </b><font color="Maroon">RR</font> = <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q2</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) </span>}</span> ;<br/>
<a NAME="E5:34"/>
<span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q2</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<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="E1:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q2</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
 ex <font color="Olive">q2</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> 2</span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q2</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">RR</font> = <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q2</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:34"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">RR</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:34"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RR</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:34"/><b>then </b>
 ex <font color="Olive">q1</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> 2</span>)</span> st <br/>( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; ( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q1</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E10:34"/>
 ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">P</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>


</div>
